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Full text of "Quantum electrodynamics"



Quantum 
Electrodynamics 



Frontiers in Physics 

A Lecture Note and Reprint Series 



DAVID PINES, Editor 



N. Bloembergen NUCLEAR MAGNETIC RELAXATION: A Re- 
print Volume 

Geoffrey F. Chew S-MATRIX THEORY OF STRONG INTER- 
ACTIONS: A Lecture Note and Reprint Volume 

R. P. Feynman QUANTUM ELECTRODYNAMICS: A Lecture 
Note and Reprint Volume 

R. P. Feynman THE THEORY OF FUNDAMENTAL PROC- 
ESSES: A Lecture Note Volume 

Hans Frauenf elder THE MOSSBAUER EFFECT: A Collection 
of Reprints with an Introduction 

David Pines THE MANY-BODY PROBLEM: A Lecture Note 
and Reprint Volume 

L. Van Hove, N. M. Hugenholtz, and L. P. Howland PROB- 
LEMS IN THE QUANTUM THEORY OF MANY-PARTICLE 
SYSTEMS 



Quantum 
Electrodynamics 



A Lecture Note and Reprint Volume 



R. P. FEYNMAN 

California Institute of Technology 

Notes corrected by 

E. R. HUGGINS 
H. T. YURA 

California Institute of Technology 



W. A. BENJAMIN, INC. 

New York 1961 



QUANTUM ELECTRODYNAMICS 
A Lecture Note and Reprint Volume 

Copyright © 1961 by W. A. Benjamin, Inc. 
All rights reserved. 

Library of Congress Catalog Card Number: 61-18179 
Manufactured in the United States of America 






c -^ 



W. A. BENJAMIN, INC. 

2465 Broadway, New York 25, New York 



EDITOR'S FOREWORD 



The problem of communicating in a coherent fashion the recent 
developments in the most exciting and active fields of physics 
seems particularly pressing today. The enormous growth in the 
number of physicists has tended to make the familiar channels of 
communication considerably less effective. It has become increas- 
ingly difficult for experts in a given field to keep up with the cur- 
rent literature; the novice can only be confused. What is needed is 
both a consistent account of a field and the presentation of a definite 
"point of view" concerning it. Formal monographs cannot meet 
such a need in a rapidly developing field, and, perhaps more im- 
portant, the review article seems to have fallen into disfavor. In- 
deed, it would seem that the people most actively engaged in devel- 
oping a given field are the people least likely to write at length 
about it. 

"Frontiers in Physics" has been conceived in an effort to im- 
prove the situation in several ways. First, to take advantage of the 
fact that the leading physicists today frequently give a series of 
lectures, a graduate seminar, or a graduate course in their special 
fields of interest. Such lectures serve to summarize the present 
status of a rapidly developing field and may well constitute the only 
coherent account available at the time. Often, notes on lectures ex- 
ist (prepared by the lecturer himself, by graduate students, or by 
postdoctoral fellows) and have been distributed in mimeographed 
form on a limited basis. One of the principal purposes of the 
"Frontiers in Physics" series is to make such notes available to 
a wider audience of physicists. 



vi > EDITOR'S FOREWORD 

It should be emphasized that lecture notes are necessarily rough 
and informal, both in style and content, and those in the series will 
prove no exception. This is as it should be. The point of the series 
is to offer new, rapid, more informal, and, it is hoped, more effec- 
tive ways for physicists to teach one another. The point is lost if 
only elegant notes qualify. 

A second way to improve communication in very active fields of 
physics is by the publication of collections of reprints of recent ar- 
ticles. Such collections are themselves useful to people working in 
the field. The value of the reprints would, however, seem much en- 
hanced if the collection would be accompanied by an introduction of 
moderate length, which would serve to tie the collection together 
and, necessarily, constitute a brief survey of the present status of 
the field. Again, it is appropriate that such an introduction be in- 
formal, in keeping with the active character of the field. 

A third possibility for the series might be called an informal 
monograph, to connote the fact that it represents an intermediate 
step between lecture notes and formal monographs. It would offer 
the author an opportunity to present his views of a field that has 
developed to the point at which a summation might prove extraor- 
dinarily fruitful, but for which a formal monograph might not be 
feasible or desirable. 

Fourth, there are the contemporary classics— papers or lectures 
which constitute a particularly valuable approach to the teaching 
and learning of physics today. Here one thinks of fields that lie at 
the heart of much of present-day research, but whose essentials 
are by now well understood, such as quantum electrodynamics or 
magnetic resonance. In such fields some of the best pedagogical 
material is not readily available, either because it consists of pa- 
pers long out of print or lectures that have never been published. 

"Frontiers in Physics" is designed to be flexible in editorial 
format. Authors are encouraged to use as many of the foregoing 
approaches as seem desirable for the project at hand. The publish- 
ing format for the series is in keeping with its intentions. Photo- 
offset printing is used throughout, and the books are paperbound, in 
order to speed publication and reduce costs. It is hoped that the 
books will thereby be within the financial reach of graduate students 
in this country and abroad. 

Finally, because the series represents something of an experi- 
ment on the part of the editor and the publisher, suggestions from 
interested readers as to format, contributors, and contributions 
will be most welcome. 



DAVID PINES 
Urbayia, Illinois 
August 1961 



PREFACE 



The text material herein constitutes notes on the third of a three -semes- 
ter course in quantum mechanics given at the California Institute of Tech- 
nology in 1953. Actually, some questions involving the interaction of light 
and matter were discussed during the preceding semester. These are also 
included, as the first six lectures. The relativistic theory begins in the 
seventh lecture. 

The aim was to present the main results and calculational procedures of 
quantum electrodynamics in as simple and straightforward a way as possi- 
ble. Many of the students working for degrees in experimental physics did 
not intend to take more advanced graduate courses in theoretical physics. 
The course was designed with their needs in mind. It was hoped that they 
would learn how one obtains the various cross sections for photon processes 
which are so important in the design of high-energy experiments, such as 
with the synchrotron at Cal Tech. For this reason little attention is given to 
many aspects of quantum electrodynamics which would be of use for theo- 
retical physicists tackling the more complicated problems of the interaction 
of pions and nucleons. That is, the relations among the many different for- 
mulations of quantum electrodynamics, including operator representations 
of fields, explicit discussion of properties of the S matrix, etc., are not in- 
cluded. These were available in a more advanced course in quantum field 
theory. Nevertheless, this course is complete in itself, in much the way that 
a course dealing with Newton's laws can be a complete discussion of me- 
chanics in a physical sense although topics such as least action or Hamilton's 
equations are omitted. 

The attempt to teach elementary quantum mechanics and quantum elec- 
trodynamics together in just one year was an experiment. It was based on 
the idea that, as new fields of physics are opened up, students must work 



VII 



viii ^ PREFACE 



their way further back, to earlier stages of the educational program. The 
first two terms were the usual quantum mechanical course using Schiff 
(McGraw-Hill) as a main reference (omitting Chapters X, XII, XIII, and XIV, 
relating to quantum electrodynamics). However, in order to ease the tran- 
sition to the latter part of the course, the theory of propagation and potential 
scattering was developed in detail in the way outlined in Eqs. 15-3 to 15-5. 
One other unusual point was made, namely, that the nonrelativistic Pauli 
equation could be written as on page 6 of the notes. 

The experiment was unsuccessful. The total material was too much for 
one year, and much of the material in these notes is now given after a full 
year graduate course in quantum mechanics. 

The notes were originally taken by A. R. Hibbs. They have been edited 
and corrected by H. T. Yura and E. R. Huggins. 

R. P. FEYNMAN 

Pasadena, California 
November 1961 



CONTENTS 

Editor's Foreword v 

Preface vii 

Quantum Electrodynamics 1 

Interaction of Light with Matter— Quantum Electrodynamics 3 

Resume of the Principles and Results of Special Relativity 23 

Relativistic Wave Equation 34 

Solution of the Dirac Equation for a Free Particle 56 

Potential Problems in Quantum Electrodynamics 71 

Relativistic Treatment of the Interaction of Particles with 

Light 91 

Interaction of Several Electrons 118 

Discussion and Interpretation of Various "Correction" Terms 128 

Pauli Principle and the Dirac Equation 162 

Summary of Numerical Factors for Transition Probabilities • 

Phys. Rev., 84, 123 (1951) 165 

The Theory of Positrons • Phys. Rev., 76, 749-759 (1949) 167 

Space-Time Approach to Quantum Electrodynamics • Phys. Rev., 

76, 769-789 (1949) 178 



IX 



ACKNOWLEDGMENTS 

The publisher wishes to acknowledge the assistance of the American Institute 
of Physics in the preparation of this volume, specifically their permission to 
reprint the three articles from the Physical Review. 



Quantum 
Electrodynamics 



Interaction 
of Light with Matter- 
Quantum 
Electrodynamics 

First Lecture 

The theory of interaction of light with matter is called quantum electro- 
dynamics. The subject is made to appear more difficult than it actually is 
by the very many equivalent methods by which it may be formulated. One of 
the simplest is that of Fermi. We shall take another starting point by just 
postulating for the emission or absorption of photons. In this form it is most 
immediately applicable. 



DISCUSSION OF FERMI'S METHOD! 

Suppose all the atoms of the universe are in a box. Classically the box 
may be treated as having natural modes describable in terms of a distribu- 
tion of harmonic oscillators with coupling between the oscillators and 
matter. 

The transition to quantum electrodynamics involves merely the assump- 
tion that the oscillators are quantum mechanical instead of classical. They 
then have energies (n f l/2)hu;, n = 0, 1 ..., with zero-point energy l/2hu;. 
The box is considered to be full of photons with a distribution of energies 
nnu;. The interaction of photons with matter causes the number of photons 
of type n to increase by ±1 (emission or absorption). 

Waves in a box can be represented as plane standing waves, spherical 
waves, or plane running waves exp (iK- x). One can say there is an instan- 



tRevs. Modern Phys., 4, 87 (1932). 



QUANTUM ELECTRODYNAMICS 



taneous Coulomb interaction e 2 /r i j between all charges plus transverse 
waves only. Then the Coulomb forces may be put into the Schrodinger equa- 
tion directly. Other formal means of expression are Maxwell's equations 
in Hamiltonian form, field operators, etc. 

Fermi's technique leads to an infinite self-energy term e 2 /r i{ . It is pos- 
sible to eliminate this term in suitable coordinate systems but then the trans- 
verse waves contribute an infinity (interpretation more obscure). This anom- 
aly was one of the central problems of modern quantum electrodynamics. 



Second Lecture 

LAWS OF QUANTUM ELECTRODYNAMICS 

Without justification at this time the "laws of quantum electrodynamics" 
will be stated as follows: 

1. The amplitude that an atomic system will absorb a photon during the 
process of transition from one state to another is exactly the same as the 
amplitude that the same transition will be made under the influence of a po- 
tential equal to that of a classical electromagnetic wave representing that 
photon, provided: (a) the classical wave is normalized to represent an en- 
ergy density equal to fico times the probability per cubic centimeter of find- 
ing the photon; (b) the real classical wave is split into two complex waves 
e -iwt anc j e +iwt ) anc [ on iy the e" iwt part is kept; and (c) the potential acts 
only once in perturbation; that is, only terms to first order in the electro- 
magnetic field strength should be retained. 

Replacing the word "absorbed" by "emit" in rule 1 requires only that 
the wave represented by exp (+iu)t) be kept instead of exp (— icot). 

2. The number of states available per cubic centimeter of a given polar- 
ization is 

d 3 K/(27r) 3 

Note this is exactly the same as the number of normal modes per cubic cen- 
timeter in classical theory. 

3. Photons obey Bose- Einstein statistics. That is, the states of a collec- 
tion of identical photons must be symmetric (exchange photons, add ampli- 
tudes). Also the statistical weight of a state of n identical photons is 1 in- 
stead of the classical n! 

Thus, in general, a photon may be represented by a solution of the classi- 
cal Maxwell equations if properly normalized. 

Although many forms of expression are possible it is most convenient to 
describe the electromagnetic field in terms of plane waves. A plane wave 
can always be represented by a vector potential only (scalar potential made 
zero by suitable gauge transformation). The vector potential representing a 
real classical wave is taken as 



INTERACTION OF LIGHT WITH MATTER 



A = ae cos (cot - K-x) 



We want the normalization of A to correspond to unit probability per cu- 
bic centimeter of finding the photon. Therefore the average energy density 
should be fico. 

Now 

E= (l/c)(8A/8t) = (wa/c)e sinM -K-x) 
and 

|B| = |E| 
for a plane wave. Therefore the average energy density is equal to 



(1/8tt)(|E| 2 + |B| 2 ) = (l/47r)(w 2 a 2 /c 9 ) sin*M - K-x) 
= (l/87r)(co 2 a 2 /c 2 ) 
Setting this equal to hco we find that 



Thus 



a = V87rhc 2 /w 

A = V87rhc z /w e cos (cot - K • x) 



= v / 47rhc/2o) e {exp[-i(ojt - K«x)] + exp[+i(cut - K» x) ]} 

Hence we take the amplitude that an atomic system will absorb a photon 
to be 



V47rHc 2 /2co exp [-i(u;t - K • x)] 

For emission the vector potential is the same except for a positive exponen- 
tial. 

Example: Suppose an atom is in an excited state *j with energy E. and 
makes a transition to a final state * f with energy E f . The probability of 
transition per second is the same as the probability of transition under the 
influence of a vector potential ae exp[+i(cot - K-X)] representing the emit- 
ted photon. According to the laws of quantum mechanics (Fermi's golden 
rule) 

Trans, prob./sec = 2tt/H | f (potential) . | 2 • (density of states) 

., , K 2 dK dfi w 2 dn 

Density of states = 



(27r) J d(coh) (27rcrfi 



QUANTUM ELECTRODYNAMICS 



The matrix element U fi = | f (potential)^ 2 is to be computed from pertur- 
bation theory. This is explained in more detail in the next lecture. First, 
however, we shall note that more than one choice for the potential may give 
the same physical results. (This is to justify the possibility of always choos- 
ing = for our photon.) 



Third Lecture 

The representation of the plane-wave photon by the potentials 
A(x, t) = ae exp [-i(ojt - K • x)] 

0=0 

is essentially a choice of "gauge." The fact that a freedom of choice exists 
results from the invariance of the Pauli equation to the quantum-mechanical 
gauge transform. 

The quantum-mechanical transformation is a simple extension of the 
classical, where, if 



•V + 90/ at 



and 



B = Vx A 

and if x is an Y scalar, then the substitutions 
A = A +Vx 
0' = + 9x/9t 

leave E and B invariant. 

In quantum mechanics the additional transformation of the wave function 



is introduced. The invariance of the Pauli equation is shown as follows. The 
Pauli equation is 



n d* _ 1 
i at 2m 



(p-±Af..(,-iAJ 



^ + e0^ 



Then, since 



INTERACTION OF LIGHT WITH MATTER 



dx dx dx dx 



Vie' 1 **) =e- J Mp-Vx)* 



and 



The partial derivative with respect to time introduces a term 
(ax/3t)*e" ix , and this may be included with 0e" ix *. Therefore the sub- 
stitutions 



*' 


- e 


iX^ 


A' 


= A 


e _ 

+ -v x 


0' 


= 


+ Ox/9t) 



leave the Pauli equation unchanged. 

The vector potential A as defined for a photon enters the Pauli Hamil- 
tonian as a perturbation potential for a transition from state i to state f. 
Any time -dependent perturbation which can be written 

AH = e iwt U(x,y,z) 

results in the matrix element U fi given by 

U fi = /^♦AH^jd vol 

= f<p{*(x) exp[i(E f /R)t] e iwt U(x) exp[-i(E f /h)t] ^(x)d vol 

This expression indicates that the perturbation has the same effect as a time- 
independent perturbation U(x,y,z) between initial and final states whose en- 
ergies are, respectively, E.- w * and E f . As is well knownf the most impor- 
tant contribution will come from the states such that E f = E. -coh. 

Using the previous results, the probability of a transition per second is 

-r^ ^^ 27T |TT ,0 00 2 dfi 

P fi d = — | Uft | * w 



fSee, for example, L. D. Landau and E. M. Lifshitz, "Quantum Mechan- 
ics; Non-Relativistic Theory,'' Addison-Wesley, Reading, Massachu- 
setts, 1958, Sec. 40. 



QUANTUM ELECTRODYNAMICS 



To determine U fi , write 



H = T L fp- -A)' - ^-(g.VxA) +eV 
2m \ c / 2mc 






e 2 



2mc 2 



A- A 



Because of the rule that the potential acts only once, which is the same 
as requiring only first-order terms to enter, the term in A • A does not en- 
ter this problem. Making use of A = ae exp [— i(ajt — K • x)] and the two 
operator relations 

(1) Vx A = Kxe e + iK ' x e lu,t 

(2) pe +iK ' x = e +iK * x (p -HK) 
or 

p. ee +iK'X =e +iK« *(p. e -RKe) 

where K-e = (which follows from the choice of gauge and the Maxwell 
equations), we may write 

u fi = aj(/) f *[-(e/2mc)(p-e e + iK * x + e +iK * x e-p) 
+ (ehi/2mc) a • (K x e) e + iK * x ] <p { d vol 

This result is exact. It can be simplified by using the so-called "dipole" 
approximation. To derive this approximation consider the term 
(e/2mc)(p-e e +lK " x ), which is the order of the velocity of an electron in the 
atom, or the current. The exponent can be expanded. 

e+ iK- x = 1 + iK . x + !/ 2 (iK- x) 2 + •• • 

K- x is of the order a /A., where a = dimension of the atom and A = wave- 
length. If a /A« 1, all terms of higher order than the first in a A ma Y De 
neglected. To complete the dipole approximation, it is also necessary to 
neglect the last term. This is easily done since the last term may be taken 
as the order of (RK/mc) = (hKc/mc 2 ) « (mv 2 /2mc 2 ). Although such a term is 
negligible even this is an overestimate. More correctly, 



INTERACTION OF LIGHT WITH MATTER 

(eRi/2mc)o- • (K x e) e +iK ' x « v/c x [matrix element of 

(J • (KX p)] 
The matrix element is 

J <f> f *<r • (Kxp)0 i d vol 
A good approximation allows the separation 

f * = f *(x)U f *(spin) 
and 

0i = 0i (x)U i *(spin) 

Then to the accuracy of this approximation the integral is 

JVf*<x)0 1 (x)U f *(o-- (Kxp))Ui d vol = 

since the states are orthogonal. 

For the present, the dipole approximation is to be used. Then 

e Pfi' e 

U fi = -a 

11 cm 

where 

Pfi • e = /0f*(P-e)tf>i =e • J0 f *P0i d vol 

So 

2 



p «-ffe']*---> ,d ° 



Cc 



(2tt) 3 



Using operator algebra, p fi /m = haj fi x f . , so that 

P fl dfi = a 2 [e 2 ^ 4 /(27r) 2 ](e- x fi ) 2 dfi 

where x fi = J f * x 0j d vol. The total probability is obtained by inte- 
grating P fi over dQ, thus 

2 i. 

Total prob./sec = J a 2 ^~J (© " x fi ) 2 dfi 

= a 2 ^ i /|x fi | 2 S in 3 «d9 

Z7T 

= a 2 4e 2 co 4 |x f1 | 2 /67r 



10 



QUANTUM ELECTRODYNAMICS 



The term e • x fi is resolved by noting (Fig. 3-1) 

|x fi • e| = |x f} | sin G 



FIG. 3-1 



Substituting for a' 



Total prob./sec = - |- ^j |x fi | 2 



Fourth Lecture 



Absorption of Light. The amplitude to go from state k to state 1 in time 
T (Fig. 4-1) is given from perturbation theory by 



a lk = -U/h) 



Jo expQ-E,t)u lk (t) exp^-^Ektjdt 




FIG. 4-1 



INTERACTION OF LIGHT WITH MATTER 11 

where the time dependence of U kl (t) is indicated by writing 

U lk (t)=u, k e- lwt 

(In accord with the rules of Lecture 2, the argument of the exponential is 
minus and only terms which are linear in the potential are included.) Using 
this time dependence and performing the integration, 



[i< E >- 



exp — (Ej-nw-Ek) 



- 1 



a lk = El -hou-E k Ulk 

the transition probability is given by 

2 _ 4sin 2 (AT/2n) , 

|a ]k | - A 2 l u ik I A-E 1 -E k -Bo; 

This is the probability that a photon of frequency w traveling in direction 
(6, 0) will be absorbed. The dependence on the photon direction is contained 
in the matrix element u lk . For example, see Eq. (4-1) for the directional 
dependence in the dipole approximation. 

If the incident radiation contains a range of frequencies and directions, 
that is, suppose 

(^probability that a photon is present with fre- 
P(ct),0,0)da; df2 = J quency oj to cu + dco and in solid angle dfi 
[.about the direction (6,cp) 

and the probability of absorption of any photon traveling in the (6,<p) direc- 
tion is desired, it is necessary to integrate over all frequencies. This ab- 
sorption probability is 

r°° 4 sin 2 (AT/2h) , l2 ™ n v , n „ 
J A 2 " ' Ulkl p < w ' ^) dw dn 

when T is large, the factor (A)" 2 sin 2 (AT/2h) has an appreciable value only 
for Hco near Ej-E k , and ~P(<jo,6,(p) will be substantially constant over the 
small range in w which contributes to the integral so that it may be taken 
out of the integral. Similarly for u lk , so that 

Trans, prob. = 2 7 r(rl)- 1 |u lk | 2 P(w lk ,0,0)dfi (4-1) 

where 

ha; lk = (Ej-E k ) 






12 ^ QUANTUM ELECTRODYNAMICS 

This can also be written in terms of the incident intensity (energy crossing 
a unit area in unit time) by noting that 

Intensity = i(oj,9,(p)du dQ = hwc P(o>,0,0) dco dfi 

Thus 

Trans, prob. - 27r(h)- 1 |u lk | 2 (fico^c)- 1 i(co lk ,0,0)dfi (4-2) 

Using the dipole approximation, in which 



u ]k = V27T/W (e/mc)(p lk • e) 



= V27r/co lk (e/c)ho; lk (x lk • e) 

the total probability of absorption (per second) is 

47r 2 e 2 (hc)- 1 (x lk • e) 2 i(w lk , 6, 0) dti (4-3) 

It is evident that there is a relation between the probability of spontane- 
ous emission, with accompanying atomic transition from state 1 to state k, 

Probability of spontaneous 1 = , , , , dfi 

emission/sec J '" K11 1K 

and the absorption of a photon with accompanying atomic transition from 
state k to state 1, Eq. (4-1), although the initial and final states are re- 
versed since |u lk | = |u kl | . This relation may be stated most simply in terms 
of the concept of the probability n(oj,^,0) that a particular photon state is 
occupied. Since there are (27rc)T 3 <jo 2 du) dQ photon states in frequency range 
dco and solid angle d£2, the probability that there is some photon within this 
range is 

P(w,0,0) doo dfi = n(co,0,0)(2coc)~ 3 oj 2 da) dfi 

Expressing the probability of absorption in terms of n(o; > 0,0), 

Trans, prob. /sec = 27r(h) _1 |u lk | 2 n(a),0,0)(27rc)- 3 u; kl 2 d£3 

(4-4) 

This equation may be interpreted as follows. Since n(a;, #,</>) is the prob- 
ability that a photon state is occupied, the remainder of the terms of the 
right-hand side must be the probability per second that a photon in that state 
will be absorbed. Comparing Eq. (4-4) with the rate of spontaneous emis- 
sion shows that 



INTERACTION OF LIGHT WITH MATTER 



13 



'Prob./sec of absorption "1 
of a photon from a state >■ 
.(per photon in that state) J 



prob./sec of spontaneous 
emission of a photon into 
that state 



In what follows, it will be shown that Eq. (4-4) is correct even when there is 
a possibility of more than one photon per state provided n(co,0,0) is taken as 
the mean number of photons per state. 

If the initial state consists of two photons in the same photon state, it will 
not be possible to distinguish them and the statistical weight of the initial 
state will be 1/2! However, the amplitude for absorption will be twice that 
for one photon. Taking the statistical weight times the square of the ampli- 
tude for this process, the transition probability per second is found to be 
twice that for only one photon per photon state. When there are three pho- 
tons per initial photon state and one is absorbed, the following six processes 
(shown on Fig. 4-2) can occur. 





Any of the three incident photons may be absorbed and, in addition, there is 
the possibility that the photons which are not absorbed may be interchanged. 
The statistical weight of the initial state is 1/3 !, the statistical weight of the 
final state is 1/2! , and the amplitude for the process is 6. Thus the transi- 
tion probability is (1/3 !)(l/2!)(6) 2 = 3 times that if there were one photon 
per initial state. In general, the transition probability for n photons per 
initial photon state is n times that for a single photon per photon state, so 
Eq. (4-4) is correct if n(ou, B,(p) is taken as the mean number of photons per 
state. 



14 



QUANTUM ELECTRODYNAMICS 



A transition that results in the emission of a photon may be induced by 
incident radiation. Such a process (involving one incident photon) could be 
indicated diagrammatically, as in Fig. 4-3. 





FIG. 4-3 

One photon is incident on the atom and two indistinguishable photons come 
off. The statistical weight of the final state is 1/2 ! and the amplitude for 
the process is 2, so the probability of emission for this process is twice 
that of spontaneous emission. For n incident photons the statistical weight 
of the initial state is 1/n!, the statistical weight of the final state is 
l/(n + 1) !, and the amplitude for the process is (n + 1) ! times the amplitude 
for spontaneous emission. The probability (per second) of emission is then 
n + 1 times the probability of spontaneous emission. The n can be said to 
account for the induced part of the transition rate, while the 1 is the spon- 
taneous part of the transition rate. 

Since the potentials used in computing the transition probability have 
been normalized to one photon per cubic centimeter and the transition prob- 
ability depends on the square of the amplitude of the potential, it is clear 
that when there are n photons per photon state the correct transition prob- 
ability for absorption would be obtained by normalizing the potentials to n 
photons per cubic centimeter (amplitude yfn times as large). This is the 
basis for the validity of the so-called semiclassical theory of radiation. In 
that theory absorption is calculated as resulting from the perturbation by a 
potential normalized to the actual energy in the field, that is, to energy nfiu; 
if there are n photons. The correct transition probability for emission is 
not obtained this way, however, because it is proportional to n + 1. The er- 
ror corresponds to omitting the spontaneous part of the, transition prob- 
ability. In the semiclassical theory of radiation, the spontaneous part of the 
emission probability is arrived at by general arguments, including the fact 
that its inclusion leads to the observed Planck distribution formula. Ein- 
stein first deduced these relationships by semiclassical reasoning. 



INTERACTION OF LIGHT WITH MATTER 15 



Fifth Lecture 

Selection Rules in the Dipole Approximation. In the dipole approximation 
the appropriate matrix element is 

x if = J * f *x*j d vol 

The components of jaff x if are x if> y if , z if and 

Trans, prob. « |x if | 2 + |y if | 2 + |z if | 2 

Selection rules are determined by the conditions that cause this matrix ele- 
ment to vanish. For example, if in hydrogen the initial and final states are 
S states (spherically symmetrical), X if = and transitions between these 
states are "forbidden." For transitions from P to S states, however, 
x if £ and they are "allowed." 

In general, for single electron transitions, the selection rule is 

AL = ±1 

This may be seen from the fact that the coordinates x, y, and z are essen- 
tially the Legendre polynomial Pj. If the orbital angular momentum of the 
initial state is n, the wave function contains P n . But 

P t P n = [l/(2n+ 1)] [nP n _ t + (n+ l)P n+1 ] 

Hence for the matrix element not to vanish, the angular momentum of the 
final state must be n ± 1, so that its wave function will contain either P n+ i 
or P n _i. 

For a complex atom (more than one electron), the Hamiltonian is 

H =£)(l/2m)[P a - ( e /c)A(x a )] 2 + Coulomb terms 

a 

The transition probability is proportional to |P mn | 2 = |2(P a ) mn | 2 , where the 
sum is over all the electrons of the atom. As has been shown, (P a ) mn is the 
same, up to a constant, as (x a ) mn , and the transition probability is propor- 
tional to 

l X mn| 2 = £< X «>n 
a 

In particular, for two electrons the matrix element is 
/# f *(x 1 ,x 2 )(x 1 + x 2 )* i (x 1 ,x 2 ) dxj dx 2 

Xj + x 2 behaves under rotation of coordinates similarly to the wave function 
of some "object" with unit angular momentum. If the "object" and the atom 



16 QUANTUM ELECTRODYNAMICS 

in the initial state do not interact, then the product (x t + ^2)^1 (Xi>x 2 ) can be 
formally regarded as the wave function of a system (atom + object) having 
possible values of Jj + 1, Jj , and J. — 1 for total angular momentum. There- 
fore the matrix element is nonzero only if J f , the final angular momentum, 
has one of the three values J i ± 1 or J } . Hence the general selection rule 
AJ =±1, 0. 

Parity. Parity is the property of a wave function referring to its behavior 
upon reflection of all coordinates. That is, if 

*(-Xi, -x 2 , ....) = +*(Xi,X 2 , ...) 

parity is even; or if 

*(-Xi, -x 2 , ...) = -*(xi,x 2 , ...) 

parity is odd. 

If in the matrix elements involved in the dipole approximation one makes 
the change of variable of integration x = -x', the result is 

x if = /*f *(x)x*j(x) d 3 x = J* f *(-x , )(-x , )*o(-x / ) dV 

If the parity of * f is the same as that of *j, it follows that 

x if = -x if = 

Hence the rule that parity must change in allowed transitions. For a one- 
electron atom, L determines the parity; therefore, AL = would be forbid- 
den. In many-electron atoms, L does not determine the parity (determined 
by algebraic, not vector, sum of individual electron angular momenta), so 
AL = transitions can occur. The 0— transitions are always forbidden, 
however, since a photon always carries one unit of angular momentum. 

All wave functions have either even or odd parity. This can be seen from 
the fact that the Hamiltonian (in the absence of an external magnetic field) 
is invariant under the parity operation. Then, if H^(x) = E*(x), it is also 
true that H*(-x) = E\&(-x). Therefore, if the state is nondegenerate, it 
follows that either *(-x) = *(x) or *(-x) = -^(x). If the state is degen- 
erate, it is possible that \fr(-x) * *(x). But then a complete solution would 
be one of the linear combinations 

*(x) + ^(-x) even parity 

^(x) - *(-x) odd parity 

Forbidden Lines. Forbidden spectral lines may appear in gases if they 
are sufficiently rarefied. That is, forbiddenness is not absolute in all cases. 
It may simply mean that the lifetime of the state is much longer than if it 



INTERACTION OF LIGHT WITH MATTER 17 



were allowed, but not infinite. Thus, if the collision rate is small enough 
(collisions of the second kind ordinarily cause de-excitation in forbidden 
cases), the forbidden transition may have sufficient time to occur. 
In the nearly exact matrix element 

f* f *(e- p)e~ iK ' x *j d 3 x 

the dipole approximation replaces e" iK * x by 1. If this vanishes, the transi- 
tion is forbidden, as described in the foregoing. The next higher or quadru- 
pole approximation would then be to replace e" lK * x by 1 - i/K- x, giving 
the matrix element 

-i /# f *(e -p)(K- x)*id 3 x 

For light moving in the z direction and polarized in the x direction, this 
becomes 



-iK /* f *(p x z) *.d 3 x - -iK| f (p x z)j 



and the transition probability is proportional to 

whereas in the dipole approximation it was proportional to 

lf(Px)i| 2 

Therefore the transition probability in the quadrupole approximation is at 
least of the order of (Ka) 2 = a 2 /ft , smaller than in the dipole approxima- 
tion, where a is of the order of the size of the atom, and ft the wavelength 
emitted. 

Problem: Show that 

H(xz) - (xz)H = (h/mi)(p x z + xp z ) 
and consequently that 

[(K/mi)(p x z + xp z )] mn = (xz) mn (E m -E n ) 
Note that p x z can be written as the sum 

p x z = 1/2 (p x z + xp z ) + l/2(p x z - xp z ) 

From the preceding problem, the first part of p x z is seen to be equivalent, 
up to a constant, to xz, which behaves similarly to a wave function for angu- 



QUANTUM ELECTRODYNAMICS 



lar momentum 2, even parity. The second part is the angular momentum 
operator L , which behaves like a wave function for angular momentum 1, 
even parity. Therefore the selection rules corresponding to the first part 
are seen to be A J = ±2, +1, with no parity change. This type of radiation 
is called electric quadrupole. The selection rules for the second part of p x z 
are A J = ±1, 0, no parity change, and the corresponding radiation is called 
magnetic dipole. Note that unless A J ±2, the two types of radiation cannot 
be distinguished by the change in angular momentum or parity. If A J = ±1, 0, 
they can only be distinguished by the polarization of the radiation. Both types 
may occur simultaneously, producing interference. 

In the case of electric quadrupole radiation, it is implicit in the rules 
that 1/2 — 1/2 and 0—1 transitions are forbidden (even though A J may be 
± 1), since the required change of 2 for the vector angular momentum is im- 
possible in these cases. 

Continuing to higher approximations, it is possible by similar reasoning 
to deduce the vector change in angular momentum, or angular momentum of 
the photon, and the selection rules for parity change and change of total an- 
gular momentum A J associated with the various multipole orders (Table 5-1). 



TABLE 5-1. Classification of Transitions and Their Selection Rules 







Electric 


Magnetic 


Electric 


Magnetic 


Electric 




Multipole 


dipole 


dipole 


quadrupole quadrupole 


octupole 


^1 


Angular 


1 


1 


2 


2 


3 


A 


► momentum 












fy J 


Parity 


Odd 


Even 


Even 


Odd 


Odd 


Parity change 


Yes 


No 


No 


Yes 


Yes 


Change of total 


±1,0 


±1,0 


±2,±1,0 


±2, ±1,0 


±3,±2,±1,0| 




I angular mo- 












mentum A J 


No 0—0 


No 0—0 


No 0-0 

1 1 

2 2 


No 0-0 

1 1 

2 2 


No 0-0 

1 1 

2 2 


^ 








0—1 


0—1 


etc. (see i 












following) 1 

j 



Actually all the implicit selection rules for A J, which become numerous 
for the higher multipole orders, can be expressed explicitly by writing the 
selection rule as 



|J f -J, 



1 < J, + J 



where 2 is the multipole order or 1 is the vector change in angular mo- 
mentum. 



INTERACTION OF LIGHT WITH MATTER 19 



It turns out that in so-called parity-favored transitions, wherein the prod- 
uct of the initial and final parities is (— l)Jf " ^i and the lowest possible mul- 
tipole order is J f - Jj , the transition probabilities for multipole types con- 
tained within the dashed vertical lines in Table 5-1 are roughly equal. t In 
parity-unfavored transitions, where the parity product is (— lyf~h +1 and 
the lowest multipole order is | J f — Jj | + 1, this may not be true. 



Sixth Lecture 

Equilibrium of Radiation. If a system is in equilibrium, the relative num- 
ber of atoms per cubic centimeter in two states, say 1 and k, is given by 

Nj/N k = e- (E l- E kV kT =e - hw / kT 

according to statistical mechanics, when the energies differ by ho;. Since 
the system is in equilibrium, the number of atoms going from state k to 1 
per unit time by absorption of photons ftco must equal the number going from 
1 to k by emission. If n w photons of frequency u> are present per cubic cen- 
timeter, then probabilities of absorption are proportional to n w and proba- 
bility of emission is proportional to n + 1. Thus 

N k n w =N 1 (n a; +l) 

(n- w + l)/n w = N k /N, =e*u/ kT 

n w =l/(e*"^ T -1) 

This is the Planck black-body distribution law. 

The Scattering of Light. We discuss here the phenomena of an incident 
photon being scattered by an atom into a new direction (and possibly energy) 
(see Fig. 6-1). This may be considered as the absorption of the incoming 
photon and the emission of a new photon by the atom. The two photons taking 
part in the phenomenon are represented by the vector potentials. 

A t -(27r/o; 1 ) 1 / 2 e 1 e +i(w l t - K * x) 
A 2 = (27r/co 2 ) 1 / 2 e 2 e- i(w 2t-K-x) 

The number to be determined is the probability that an atom initially in state 
k will be left in state 1 by the action of the perturbation A = Aj + A 2 in the 



or 



t For nuclei emitting gamma rays this does not seem to be true. For an 
obscure reason the magnetic radiation predominates for each order of mul- 
tipole. 



20 



QUANTUM ELECTRODYNAMICS 



No. 2 




FIG. 6-1 

time T. This probability can be computed just as any transition probability 
with the use of A lk , where 

A lk = 6 kl expf-KEj/fiyr] - (i/h) 



x j Q exp[-i(E 1 /H)(T-t 3 )] U lk (t 3 ) exp[-i(E k /h)t 3 ] dt 3 
+ S / J t4 exp[-i(Ei/n)(T-t 4 )] 



xU ln (t 4 )exp[-i(E n /h)(t 4 -t 3 )] U nk (t 3 ) exp[-i(E k /h)t 3 ] dt 3 dt 4 
The dipole approximation is to be employed and 

U = AH = (e/2mc)(p*A) + (e 2 /mc 2 )(A'A) 

where spins are neglected. 

In each integral defining A lk , each of the two vector potentials must ap- 
pear once and only once. Thus, in the first integral the term p • A of U will 
not appear in U lk . The product A • A = (A { + A 2 ) • (A t + A 2 ) will contribute 
only its cross-product term 2AiA 2 . The second integral will have no con- 
tribution from A- A , but will be the sum of two terms. The first term con- 
tains a U ln based on p • A 2 and a U nk based on p- A t . The second has U ln 
based on p • Aj and U nk on p • A 2 . The time sequences resulting in these 
two terms can be represented schematically as shown in Fig. 6-2. 

The integral resulting from the first term will now be developed in de- 
tail. 

(P'Ai) nk =(2V^i) 1 ^ 2 (p-e 1 ) nk er lw i* 



(p-A 2 )i n = (27r/cD 2 ) 1/2 (p-e 2 ) ln e 



W2 t 



Then the resulting integral is 



INTERACTION OF LIGHT WITH MATTER 



21 



No. 1 




No. 2 



atom 





No. 1 



FIG. 6-2 



^27r/(a) 1 cc; 2 ) 1 /^p.e 2 ) ln (p.e 1 ) nk 
n. 

x / / 4 exp[-i(E,/H)(T-t 4 ) + iw 2 t 4 ] 



x expf-KE^^^-taJ-iw^] exp[-i(E k /K)t 3 ] dt 3 dt 4 

The integral is similar to the integrals considered previously with regard to 
transition probabilities, and the sum becomes 

£ 27r/(a; 1 a; 2 ) 1 / 2 (p • e^ (p • e 2 ) nk e 1 * 

n 

x [sin (T • A/fi)/(E k - E n + Kc^) -A] 

where A = (E 1 +ho) 2 - E k -ft(jo{), and the phase angle <p is independent of n. 
A term with the denominator given by (E n -Rc^- E k )(Ej +ho; 2 - E n ) has been 
neglected, since previous results show that only energies such that 
Ej + hco 2 a E k + hwi are important. The final result can be written 



Trans, prob./sec = (2tt/R)|M| 2 [co 2 2 dft 2 /(27r 3 )] 



(6-1) 



(JC 



where |M| is determined from A lk by integrating over co 2 and averaging 
over e 2 . Then the complete expression for the cross section cr is 



cr dfi 5 



2 4 a »'2 



_l_y> (P-e2)i n (P-ei)nk 
mV E k +ho; 1 -E n 



(p- ei ) ln (p>e 2 ) nk i 

E k -E n -Rc 2 + ^<°i" e »>fiik 



(6-2) 



22 



QUANTUM ELECTRODYNAMICS 



The first term under the summation comes from the "first term" pre- 
viously referred to and the second from the "second term." The last term 
in the absolute brackets comes from A- A. 

If 1 * k, the scattering is incoherent, and the result is called the "Raman 
effect." If 1 = k, the scattering is coherent. 

Further, note that if all the atoms are in the ground state and 1 *k, then 
the energy of the atom can only increase and the frequency of the light w 
can only decrease. This gives rise to "Stokes lines." The opposite effect 
gives "anti-Stokes lines." 

Suppose wi = co 2 (coherent scattering) but further Kw t is very nearly equal 
to E k -E n , where E n is some possible energy level of the atom. Then one 
term in the sum over n becomes extremely large and dominates the remain- 
der. The result is called "resonance scattering." If a is plotted against w, 
then at such values of w the cross section has a sharp maximum (see 
Fig. 6-3). 



FIG. 6-3 

The "index" of refraction of a gas can be obtained by our scattering for- 
mula. It can be obtained, as for other types of scattering, by considering the 
light scattered in the forward direction. 

Self-Energy. Another phenomenon that must be considered in quantum 
electrodynamics is the possibility of an atom emitting a photon and reabsorb- 
ing the same photon. This affects the diagonal element A kk . Its effect is 
equivalent to a shift of energy of the level. One finds 

(P-e) kn (p-e) nk d 3 K 2tt 



e = e/ 



E k -E n -u> (27rh) 3 co 

where e is the direction of polarization. This integral diverges. A more 
exact relativistic calculation also gives a divergent integral. This means 
that our formulation of electromagnetic effects is not really a completely 
satisfactory theory. The modifications required to avoid this difficulty of 
the infinite self-energy will be discussed later. The net result is a very 
small shift AE in position of energy levels. This shift has been observed 
by Lamb and Rutherford. 



Resume of 
the Principles 
and Results of 
Special Relativity 

Seventh Lecture 

The principle of relativity is the principle that all physical phenomena 
would appear to be exactly the same if all the objects concerned were mov- 
ing uniformly together at velocity v; that is, no experiments made entirely 
inside of a closed spaceship moving uniformly at velocity v (relative to the 
center of gravity of the matter in the universe, for example) can determine 
this velocity. The principle has been verified experimentally. Newton's 
laws satisfy this principle; for they are unchanged when subject to a Galli- 
lean transformation, 

x' = x — vt y' = y z' = z t' = t 

because they involve only second derivatives. The Maxwell equations are 
changed, however, when subjected to this transformation, and early workers 
in this field attempted to make an absolute determination of velocity of the 
earth using this feature (Michelson-Morley experiment). Failure to detect 
any effects of this type ultimately led to Einstein's postulate that the Max- 
well equations are of the same form in any coordinate system; and, in par- 
ticular, that the velocity of light is the same in all coordinate systems. The 
transformation between coordinate systems which leaves the Maxwell equa- 
tions invariant is the Lorentz transformation: 

x cosh u - ct sinh u 



Vl-(v 2 /c 2 ) 



23 



24 QUANTUM ELECTRODYNAMICS 

y =y 



t -(xv/c 2 ) X . , _/, . 

■ - — sinh u + t cosh u 

/l - (v 2 /c 2 ) c 



where tanh u = v/c. Henceforth we shall use time units so that the speed of 
light c is unity. The latter form is written to demonstrate the analogy with 
rotation of axes, 

x 7 = x cos 9 + y sin 9 

y' = —x sin 9 + y cos 9 

Successive transformations v t and v 2 or u A and u 2 add in the sense that a 
single transformation v 3 or u 3 will give the same final system if 

v 3 - v'i + v 2 or tanh u 3 = tanh (u t + u 2 ) 

Einstein postulated (theory of special relativity) that the Newton laws must 
be modified in such a way that they, too, are unchanged in form under a 
Lorentz transformation. 

An interesting consequence of the Lorentz transformation is that clocks 
appear to run slower in moving systems; that is called time dilation. In 
transforming from one coordinate system to another it is convenient to use 
tensor analysis. To this end, a four-vector will be defined as a set of four 
quantities that transforms in the same way as x,y,z and ct. The subscript 
\i will be used to designate which of the four components is being considered; 
for example, 

Xj = x x 2 = y x 3 = z x 4 = t 

The following quantities are four-vectors: 

a d d d ■ . 

— — ~ , ~ t~ "j > + Z~ (V) four- dimensional gradient 

dx dy dz at l V & 

j x > j y . j z > P (jju) current (and charge) density 

A x , A y , A z , cp (A ) vector (and scalar) potential 

P x » Py P z » ^ (pj momentum and total energy t 



2 

|The energy E, here, is the total energy including the rest energy mc . 



SPECIAL RELATIVITY 25 



An invariant is a quantity that does not change under a Lorentz transforma- 
tion. If a and b are two four-vectors, the "product" 

a ' b -S a fi b M ~ a 4 b 4 " a l b l " a 2 b 2 - a 3 b 3 
n 

is an invariant. To avoid writing the summation symbol, the following sum- 
mation convention will be used. When the same index occurs twice, sum 
over it, placing minus in front of first, second, and third components. The 
Lorentz invariance of the continuity equation is easily demonstrated by writ- 
ing it as a "product" of four-vectors V and j : 

d P aj x aj y aj z 
v M j M =v 4 j 4 -y 1J1 -v 2 j 2 -v 3 J3=- + — + ^r+^ 

Conservation of charge in all systems if it is conserved in one system is a 
consequence of the invariance of this "product," the four-dimensional di- 
vergence V- j. Another invariant is 

P ll P u =P-P = E 2 - p x 2 - p y 2 - p z 2 = E 2 - p 2 = m 2 

(E = total energy, m = rest mass, mc 2 = rest energy, p = momentum.) Thus, 

E 2 = p 2 c 2 + m 2 c 4 

It is also interesting to note that the phase of a free particle wave function 
exp [(-i/h)(Et-p-X)] is invariant since 



Et - p-X = Et - p x x - py - p z z =p p 



/j^M 



The invariance of p^p^ can be used to facilitate converting laboratory en- 
ergies to center-of-mass energies (Fig. 6-4) in the following way (consider 
identical particles, for simplicity): 



E iab E E 

Or 9 c^ ^^ 



moving stationary 

particle particle 

Laboratory system Center-of-mass system 

FIG. 6-4 



26 



QUANTUM ELECTRODYNAMICS 



PmP u = E ubm = E +p ' 



U^U 



but 



P<> = E, 



m 



so 



E lab m = 2E 2 - m 2 



and 



E o = [im(E lab + m)] 



1/2 



The equations of electrodynamics B = V X A and E = - (l/c)(a A/at) - V 
are easily written in tensor notation, 

B x =8A z /8y - d\/dz = -V y A z +V z A y 

B y - 3A x /az - dA z /dx = - V Z A X +V X A Z 

B z ^aAy/ax- 9A x /ay = -V x A y +V y A x 

E x = - 9A x /9t - a (p/dx = -v t A x + V x A t 

E y = -8A /8t - 80/ay = -V t A y + V y A t 



aA z /at - a^/az 



•V t A z + V z A t 



where use is made of the fact that is the fourth component of the four- 
vector potential A . From the foregoing it can be seen that B x , B y , B z , E x , 
E y , and E z are the components of a second-rank tensor: 



F = V A - V A 



(7-D 



This tensor is antisymmetric (F = -F y ) and the diagonal terms (jj. - v) 
are zero; thus there are only six independent components (three components 
of E and three components of B) instead of sixteen. 








"B z 


B Y 


E 




B 7 





"B x 


E 


F P, - 


" B Y 


B x 





E 




"Ex 


- E Y 


-E z 






The Maxwell equations Vx B = 4?r J + (9E/8t) and V-E = 4irp are written 



V M F u, =4^ 



(7-2) 



SPECIAL RELATIVITY 27 



where v = 1, 2, 3, 4, that is, jj = j x , j 2 = j y , J3 = j z , J4 = p, and /i is a 
dummy index of summation. The v - 1, 2, and 3 gives the three components 
of the curl equation, and v - 4 gives the divergence equation. 

The equation satisfied by the potential A^ is found, by substituting Eq. 
(7-1) into Eq. (7-2), to be 

VVA — V V A = 47ri 
The potential A y is not unique, however, since the potential 

A/ M =A M +V M X ( 7 "3) 

(X = any scalar function of position) also satisfies this relation. Such a 
change or transformation of potential is called a gauge transformation (for 
historical reasons). We shall make the potentials more definite by assum- 
ing that all potentials have been transformed so as to satisfy the so-called 
Lorentz condition! 

V M A M =0 (7-4) 

This is convenient, because it simplifies the equation for A„ to 

(V-V)A y =47rj y (7-5) 

since V-V = ^n^u » which can be recognized as the wave equations 

V 2 A - a 2 A/9t 2 - -4ttj (7-5') 

V 2 0- d 2 (p/dt 2 = -47rp 

Sometimes Eq. (7-5') is written n 2 A M = -47rj M (D 2 = D'Alembertian opera- 
tor = V 2 - O/at) 2 = -V-V). This choice of gauge (V^ A^ = 0) is the usual 
one made in classical electrodynamics, 

V- A -30/at = (7-4') 



t This is not sufficient to completely define A. We may still use any x 
such that D 2 x = 0. 



28 ^ QUANTUM ELECTRODYNAMICS 

Eighth Lecture 

SOLUTION OF THE MAXWELL EQUATION IN EMPTY SPACE 

In empty space the plane wave solution of the wave equation 

□ 2 A M =-4ttj m =0 
is 

where e and k are constant vectors, and k„ is subject to the condition 
that 

k^k^ =k-k =0 

This may be seen from the fact that V v operating on e" lk ' x has the effect 
of multiplying by ik y (V y does not operate on e. since the coordinates are 
rectangular). Thus, 

-D 2 A, =V>„A,)=V„<-ie M k 1 ,e- il '; x ) 

= -^(k„ye- ik ' X 

Note that in these operations V y A actually forms a second-rank tensor, 
V y (V y A ) a third-rank tensor, and then contraction on the index v yields a 
first-rank tensor or vector. 

The k„ is the propagation vector with components 

k^ = co, K x , K y , K z = co, K 

so that in ordinary notation 

exp (-ik • x) = exp [-i (cot - K • X)] 

and the condition k • k = means 

co 2 - K-K = 

Problem: Show that the Lorentz condition 

V p A M = 

implies that k • e =0. 

When working in three dimensions it is customary to take the polariza- 
tion vector e such that K -e = and to let the scalar potential 0=0. But 



SPECIAL RELATIVITY 29 



this is not a unique condition; that is, it is not relativistically invariant and 
will be true only in a one-coordinate system. This would seem to be a para- 
dox attaching some uniqueness to the system in which K • e = 0, a situation 
incompatible with relativity theory. The "paradox," however, is resolved 
by the fact that one can always make a so-called gauge transformation, 
which leaves the field F^ unaltered but which does change e. Therefore, 
choosing K-e = in a particular system amounts to selecting the certain 
gauge. 

The gauge transformation, Eq. (7-3), is 

A' = A + V X 

0' =0 +Ox/9t) 

where x is a scalar. But V • A = 0, the Lorentz condition, Eq. (7-4), will 
still hold if 

V • A' =V-A + V-x =0 
or if 

□ 2 x =0 
This equation has a solution x = ae*" ik ' X , so 

A M = A m + V a e_lk ' X ) =(e p +ak M )e- ik '* 
where a is an arbitrary constant. Therefore, 

is the new polarization vector obtained by gauge transformation. In ordinary 
notation 

e' =e + aK 

ej = e 4 + aoo 

Thus, no matter what coordinate system is used, 

K-e' = K • e + a K • K = K • e + aoo 2 

can be made to vanish by choice of the constant a . 

Clearly the field is left unchanged by a gauge transformation for 

F W = v „ K ~ \ A '» = V M A„ + V,V„ X - \A V - V„V p x = F„„ 



30 QUANTUM ELECTRODYNAMICS 



the V^V^X = ^V^x because the order of differentiations is immaterial. 

RELATIVISTIC PARTICLE MECHANICS 

The components of ordinary velocity do not transform in such a manner 
that they can be components of a four-vector. But another quantity 



dz/ds = dt/ds, dx/ds, dy/ds, dz/ds 



where 



dz = dt, dx, dy, dz 

is an element of path of the particle and ds is the proper time defined by 

ds 2 = dt 2 - dx 2 - dy 2 - dz 2 

is a four- vector and is called the four-velocity u . Dividing ds 2 by dt 2 
gives the relation between proper time and local time to be 

(ds/dt) 2 = 1 - v 2 
The components of ordinary velocity are related as follows: 

dx/ds = (dx/dt) (dt/ds) = v x /(l -y 2 ) 1/2 

dy/ds =v y /(l-v 2 ) 1 / 2 

dz/ds =v z /(l-v 2 ) 1 / 2 

dt/ds = 1/(1 -v 2 ) 1 / 2 
It is evident that u„u„ = 1, for 

1 v x 2 v y 2 v z 2 1 - v 2 

u m u m = tt^2 - yzt^ - i _ v 2 " T-V = T^7 = 1 

The four-momentum is defined 

P M =mu M =m/(l- v 2 ) 1/2 , mv x /(l-v 2 ) 1/2 , mv y /(l - v 2 ) 1/2 , 

mv z /(l - v 2 ) 1/2 

Note that p 4 = m/(l - v 2 ) 1/2 is the total energy E, so that in ordinary nota- 
tion the momentum P is given by 



SPECIAL RELATIVITY 31 



P = Ev 

where v is the ordinary velocity. 

Like the velocity, the components of ordinary force defined by d/dt (mo- 
mentum) cannot form the components of a four-vector. But the quantity 



f„=dP M /ds 

does form a four-vector with the components 



f u = d/dt(mv M /VI "-' v 2 ) dt/ds = F^/Jl - v2 M = 1, 2, 3 

where F is the ordinary force. The fourth component is 



power _ rate of change of energy _ d/dt(m/Vl - v 2 ) 

4 ~ vr 1- ^ " vi - v 2 vi - v2 _ 

This is seen from the fact that m/Vl - v 2 is the total energy and also from 
the ordinary identity 



Power = F • V 



_d mV 1 
dt VI - v 2 J 



ni v 2 1 dv 2 

2 (l-v 2 ) 3/2 + (1-v 2 ) 1 / 2 dt 



mv dv 



( l_ v 2)3/2 dt dt jj_ yl 

Thus the relativistic analogue of the Newton equations is 

d/ds (p^) = f M = m d 2 z M /ds 2 (8-1) 

The ordinary Lorentz force is 

F = e(E + v X B) (8-2) 

and the rate of change of energy is 

F-v = eE-v 
Then from the preceding definition of four-force, 

f = e/(l - v 2 ) 1/2 (E + vxB) 
and 



32 QUANTUM ELECTRODYNAMICS 

f 4 = e/(l - v 2 ) 1/2 E-v 

Problem: Show that the expressions just given for f and f 4 are 
equivalent to 

f = eu F 
so that the relativistic analogue of the Newton equation becomes 

m d 2 z p /ds 2 = e(dz y /ds) F^ (8-3) 

Also show that this implies 

d/ds[(dz^/ds) 2 ] =0 
In ordinary terms the equation of motion is 

d/dt(mv/Vl - v 2 ~) = e(E + v X B) (8-4) 

It can be shown by direct application of the Lagrange equations 

d/dtOL/3v J[i )-OL/ax M ) = 
that the Lagrangian 



L = -m/l - v 2 - e0+ eA- v (8-5) 

leads to these equations of motion. Also the momenta conjugate to x is 
given by 9 L/8v or 

P =mv/(l - v 2 ) 1/2 + eA 

The corresponding Hamiltonian is 

H = e<p + [(P - eA) 2 + m 2 ] 1/2 (8-6) 

which satisfies (H-e0) 2 - (P-eA) 2 = m 2 . It is difficult to convert the 
Hamiltonian idea to a covariant or four-dimensional formulation. But the 
principle of least action, which states that the action 

S = j L dt 

shall be a minimum, will lead to the relativistic form of the equations of 
motion directly when expressed as 



SPECIAL RELATIVITY 33 

S = Jl dt = m /ds + e J A^ (dz p /ds) ds 

= J[m(dz/da • dz/da) 1/2 + eA^ dz^/da] da 
Note that by definition 

(ds/da) 2 = {dz ll /da){dz u /da) 
It is interesting that another "action, " defined 

S' =m/2 j (dz^/da) 2 da + e j A jl (z jl )(dz jI /da) da 

leads to the same result as for S in the foregoing. 

Problems: (1) Show that the Lagrangian, Eq. (8-5), leads to the 
equations of motion, Eq. (8-4), and that the corresponding Hamiltonian 
is Eq. (8-6). Also find the expression for P. (2) Show that 6S = (va- 
riation of S), where S is the action just given, leads to the same equa- 
tions . 



Relativistic 
Wave Equation 



Ninth Lecture 

UNITS 

The following convention will be used hereafter. We define the units of 
mass and time and length such that 

c = 1 (c = 2.99 793 x 10 10 cm/sec) 

R = 1 (h = 1.0544 x 10~ 27 erg/sec) 

Table 9-1 (top of page 35) is given as a useful reference for conversion to 
customary units. 

The following numerical values are useful: 

M p = mass of proton - 1836.1 m = 938.2 Mev 

Mass unit of atomic weights = 931.2 Mev 

M H = Mass of hydrogen atom = 1.00815 mass units 

M N = Mass of neutron = 784 kev + M H 

kT = 1 ev when T = 11,606° K 

N a = Avogadro's number = 6.025 x 10 23 

N.e = 96,520 coulombs 



KLEIN-GORDON, PAULI, AND DIRAC EQUATIONS 

According to relativistic classical mechanics, the Hamiltonian is given by 



H = V(p - eA) 2 + m 2 + e0 (9-1) 



34 



RELATIVISTIC WAVE EQUATION 35 



TABLE 9-1. Notations and Units 



Present 




Customary 




notation 


Meaning 


notation 


Value 


m 


Mass of electron 


m 






Energy 


mc 2 


510.99 kev 




Momentum 


mc 


1704 gauss cm 




Frequency 


mc 2 /K 






Wave number 


mc/fi 




1/m 


Length (Comp- 
ton wave- 
length)/27r 


fi/mc 


3.8615 x 10 -11 cm 




Time 


H/mc 2 




e 2 


Fine -structure 
constant 
(dimensionless) 


e 2 /hc 


1/137.038 


e 2 /m 


Classical radius 
of the electron 


e 2 /mc 2 


2.8176 x lO'^cm 


1/me 2 


Bohr radius 


a = H 2 /me 2 


0.52945 A 



If the quantum -mechanical operator -iV is used for p, the operation deter- 
mined by the square root is undefined. Thus the relativistic quantum- 
mechanical Hamiltonian has not been obtained directly from the classical 
equation, Eq. (9-1). However, it is possible to define the square of the oper- 
ator and to write 

(H -e0) 2 - (p- eA) 2 = m 2 

Then, if H = i8/9t, 

[-(fi/i)8/8t - e0] 2 *-[(R/i)(a/ax) - e/cA x ] 2 * - ... = m 2 * 

(9-2) 

where the square of an operator is evaluated by ordinary operator algebra. 
This equation was first discovered by Schrodinger as a possible relativistic 
equation. It is usually referred to as the Klein-Gordon equation. In relativ- 
istic notation it is 

(iV^ - eA M )(iV M - eA^)* = m 2 * (9-2') 



36 QUANTUM ELECTRODYNAMICS 



This equation does not allow for "spin" and therefore fails to describe 
the fine structure of the hydrogen spectrum. It is proposed now for applica- 
tion to the it meson, a particle with no spin. To demonstrate its application 
to the hydrogen atom, let A = and = -Ze/r, then let * = x (r) exp (-iEt). 
Then the equation is 

(E - Ze 2 /r) 2 x + V 2 X = m 2 X 

Let E = m + W, where W« m, and substituting V = Ze 2 /r, 

(W - V)x + V 2 x /2m = -(W - V) 2 x/2m 

Neglecting the term on the right in comparison with the first term on the 
left gives the ordinary Schrodinger equation. By using (W - V) 2 /2m as a 
perturbation potential, the student should obtain the fine-structure splitting 
for hydrogen and compare with the correct values. 

Exercise; For the Klein-Gordon equation, let 

p = i(ty*dty/dt - * 8 * */ 9 1) — e0*** = charge density 
j = — i(^*V^ — #V^*) — eA^^* = current density 

Then show (p, j) is a four-vector and show V j = 0. 

The Klein-Gordon equation leads to a result that seemed so unreasonable 
at the time it was first brought to light that it was considered a valid basis 
for rejecting the equation. This result is the possibility of negative energy 
states. To see that the Klein-Gordon equation predicts such energy states, 
consider the equation for a free particle, which can be written 

D 2 ^ = m 2 * 

where D 2 is the D'Alembertian operator. In four-vector notation, this equa- 
tion has the solution V = A exp(-ip p x M ), where p^Py = m 2 . Then, since 

P^P M =P 4 P4 ~ PxPx" PyPy " PzPz = E 2 - p • p 



there results 



E =±(m 2 + p-p) 1/2 



The apparent impossibility of negative values of E led Dirac to the de- 
velopment of a new relativistic wave equation. The Dirac equation proves to 
be correct in predicting the energy levels of the hydrogen atom and is the 
accepted description of the electron. However, contrary to Dirac's original 



RELATIVISTIC WAVE EQUATION 37 

intent, his equation also leads to the existence of negative energy levels, 
which by now have been satisfactorily interpreted. Those of the Klein- 
Gordon equation can also be interpreted. 

Exercise: Show if ^ = exp(-iEt)x (x,y,z) is a solution of the Klein- 
Gordon equation with constant A and 0, then ^ = exp(+iEt)x * is a so- 
lution with -A and -<fi replacing A and 0. This indicates one manner in 
which "negative" energy solutions can be interpreted. It is the solution 
for a particle of opposite charge to the electron, but the same mass. 

Instead of following the original method in the development of the Dirac 
equation, a different approach will be used here. The Klein-Gordon equation 
is actually the four- vector form of the Schrodinger equation. With an anal- 
ogous point of view, the Dirac equation can be developed as the four-vector 
form of the Pauli equation. 

In following such a procedure, the terms involving "spin" will be included 
in the relativistic equation. The idea of spin was first introduced by Pauli, 
but it was not at first clear why the magnetic moment of the electron had to 
be taken as he/2mc. This value did seem to follow naturally from the Dirac 
equation, and it is often stated that only the Dirac equation produces as a 
consequence the correct value of the electron's magnetic moment. However, 
this is not true, as further work on the Pauli equation showed that the same 
value follows just as naturally, i.e., as the value that produces the greatest 
simplification. Because spin is present in the Dirac equation, and absent in 
the Klein-Gordon, and because the Klein-Gordon equation was thought to be 
invalid, it is often stated that spin is a relativistic requirement. This is in- 
correct, since the Klein-Gordon equation is a valid relativistic equation for 
particles without spin. 

Thus the Schrodinger equation is 

H# = E* 
where 

H = l/2m(-iV- eA) 2 + e0 

and the Klein-Gordon equation is 

[(H - e<p) 2 - (-iV - eA) 2 ] * = m 2 * (9-3) 

Now the Pauli equation is also H\fr = E\£, where 

H = (l/2m)[a ■ (-iV- eA)] 2 + e0 (9-4) 

Thus ( -iV — eA) 2 appearing in the Schrodinger equation has been replaced 
by [o - • (— iV— eA)] 2 . Then a possible relativistic version of the Pauli equa- 
tion, in analogy to the Klein-Gordon equation, might be 



38 QUANTUM ELECTRODYNAMICS 

(H - e0) 2 * -{a- [(fi/i)V- (e/c)A]} 2 * = m 2 * 

Actually, this is incorrect, but a very similar form [with H replaced by 
i(9/at)] is correct, namely, 

[iO/8t) - e0 - <r • (-iV- eA)] 

x [iO/at) - e0 + <r • (-i V- eA)] * = m 2 * (9-5) 

This is one form of the Dirac equation. 

The wave function ^ on which the operations are being carried out is 
actually a matrix: 



•■-© 



A form closer to that originally proposed by Dirac may be obtained as 
follows. For convenience, write 

I(8/8't) - e0 = tt 4 

-iV - (e/c)A = ir 

Now let the function x be defined by (ir i + cr • ir) * = mx . 

Then Eq. (9-5) implies (7r 4 - a • tt)x = m*. This pair of equations can be 
rewritten (only to arrive at a particular conventional form) by writing 

X + * = * a 

X - * = % 
Then adding and subtracting the pair of equations for *,x , there results 



-7r 4 * b + q • 7T ^ a = m* b (9-6) 

These two equations may be written as one by employing a particular 
convention. Define a new matrix wave function as 



* = T a 2 (9-7) 




where the matrix character of * a and * b has been shown explicitly, i.e. 
actually 



I 



RELATIVISTIC WAVE EQUATION 



39 



•-(fc) «-(fc) 



Then, if the auxiliary definitions are made, 



74 = 



{Note: An example of the latter definition is 
/ 1* 



I 1 ° 

1 


\ 

o o I 


v° ° 

\0 


- "J 

-1/ 







I 



I <7 



(9-8) 




0i 



V 









1 












1 





-1 








since 


^x = 


V 





1 








o 











y y and y z are similar.) The two equations in \fr a and * b can be written as 
one in the form 

y 4 7r 4 ^ - y • 7r* = m* 

which is actually four equations in four wave functions. Then using four- 
vector notation, the Dirac equation is 



y^^ = m* 



or 



y. M (iV M - eA u )* =m* 



(9-9) 



Exercise: Show 



rnTy + r y r, 



if /x * y 

2 if /* = i; = 4 

-2 if v =M = 1, 2, 3 



that is, show 



Vt =1 

VtTx =i -YxV t 



y x 2 = y y 2 = y z 2 = -i 



'x'y ^vYx 



etc. 



40 QUANTUM ELECTRODYNAMICS 



A similar form for the Dirac equation might be obtained by a different 
argument, by comparison to the Klein-Gordon equation. Thus with 
H = i(9/9t) = i V 4 and with ecp = eA 4 , Eq. (9-3) becomes 

(iV M - eA^) 2 * = m 2 # (9-10) 

in four-vector notation. Using a similar notation in the Pauli equation, Eq. 
(9-4), but also using a = y and setting a 4 = y 4 arbitrarily (to complete the 
definition of a four-vector form of a), Eq. (9-4) can be written in a form 
similar to Eq. (9-10), 

{y M l(R/i)V^ - (e/c)A^]} 2 M/ = m 2 * (9-11) 

This should be compared to Eq. (9-9). 

Now the Pauli equation, Eq. (9-4), differs from the Schrodinger equation 
in the replacement of the three-dimensional scalar product (p - eA) 2 by the 
square of a single quantity cr • (p - eA). Analogously one might guess that 
the four-vector product (p^ — eA ) 2 in Eq. (9-10) must be replaced by the 
square of a single quantity y (p — eA ), where we must invent four ma- 
trices y„ in four dimensions in analogy to the three matrices a in three 
dimensions. The resulting equation, 



[y M (iV M -eA^)] 2 * =m 2 * (9-11) 

is essentially equivalent to Eq. (9-9) (operate on both sides of Eq. (9-9) by 
y M (iV^ - eA^) and use Eq. (9-9) again to simplify the right-hand side). 

Exercise: Show that Eq. (9-11) is equivalent to 

(iV M - eA p ) 2 - 1 ey^F^ * = m 2 * 



Tenth Lecture 

ALGEBRA OF THE y MATRICES 

In the preceding lecture the Dirac equation, 

y M (iV M - eA M )* =m* (10-1) 

was obtained, together with a special representation for the y's, 



RELATIVISTIC WAVE EQUATION 41 



where each element in these four-by-four matrices is another two-by-two 
matrix, that is, 

1 = (o 1) unit matrix °* = (1 o) etc ' 

The best way to define the y 's, however, is to give their commutation re- 
lationships, since this is all that is important in their use. The commutation 
relationships do not determine a unique representation for the y 's, and the 
foregoing is only one of many possible representations. The commutation 
relationships are 

VtTx.y, z + rx,y, Z Tt =0 ( 10 " 3 ) 

TxTy + TyTx = TxTz + T Z Tx = ° TyTz + TzTy = 

or, in a unified notation, 






6..,. =0 fji * v 



= +1 \x = v = 4 

= -1 // = v = 1,2,3 

Note that with this definition of 6„ y and the rule for forming a scalar prod- 
uct, 

Other new matrices may arise by forming products of the matrices al- 
ready defined. For example, the matrices of Eq. (10-5) are products of y's 
taken two at a time. The matrices 

T x r y T x y z y y y z y x y t y y y t y z y t 

are all independent of y x , y y , y 7 , y t . (They cannot be formed by a linear 
combination of the latter.) Similarly, products of three matrices, 

y x y y y z (= y^y t ) 
y y y z y t (=- y x ys) 
y z y t y x (= - y y ys) 
y t y x y y (=- y z ys) 



42 QUANTUM ELECTRODYNAMICS 



These are the only new products of three. For, if two of the matrices 
were equal, the product could be reduced, thus y t y y y t = — TtTt^y = - Yy 
The only new product of four that can be formed is given a special name, y 5 , 

Products of more than four must contain two equal so that they can be re- 
duced. There are, therefore, sixteen linearly independent quantities. Linear 
combinations of them may involve sixteen arbitrary constants. This agrees 
with the fact that such a combination can be expressed by a four-by-four ma- 
trix. (It is mathematically interesting then that all four-by-four matrices 
can be expressed in the algebra of the y 's; this is called a Clifford algebra 
or hypercomplex algebra. A simpler example is that of two-by-two matrices, 
the so-called algebra of quaternions, which is the algebra of the Pauli spin 
matrices.) 



Exercise: Verify that 
iTxTy 



(?.y -V* -c-y »■'.- (?:>« 



and that 



TtYx,y,z = ( n X ' y,Z ) ~ a ( defmition of &) 

\ °x y z / 



y 
It is convenient to define another y matrix, since it occurs frequently: 

(10-6) 







75 = 




3 ') 
1 0/ 




Verify 


that 














Wt 


<: -;) 


Wx,y,z 


-( 






T5 2 


= - 1 Wn 


+ y M T5 = o 




For later 


use, 


it will be convenient to define 








* = 


a ^ M - a t"Kt " 


a xYx - a yTy 


- a zTz 


from which it 


can be shown that 













= -#£ + 2a-b 


(a-b = 


a M b p ) 






a 2 


- a , a p 







K,y,z ) 



(10-7) 



(10-8) 



*y 5 = -y 5 * (io-9) 



RELATIVISTIC WAVE EQUATION 43 

For example, the first may be verified by writing 

= (a t y t -a x y x -a y y y -a z y z )(b t y t -b x y x -b y y y -b z y z ) 

and, moving the second factor to the front, by using the commutation rela- 
tionships. Doing this with the first term, (b t y t ) of the second factor produces 

b tVt( a tVt + a xT x + a y y y + a 2 y z ) 

since y t commutes with itself and anticommutes with y x , y y , and y z . By 
performing this operation on all terms, one obtains 

= b t y t [(-a t y t + a x y x + a y y y + a z y z ) + 2a t y t ] 

+ b xrx[(a t 7t - a xT x - VV - a z y z ) + 2a xy x ] 

+ b y y y [(a t y t - a x y x - a y y y - a z y z ) + 2a y y y ] 
+ b z y z [(a t y t - a x y x - a^ - a z y z ) + 2a z y z ] 
= -W + 2(b t a t y t 2 + b x a x y x 2 + b y a y y y 2 + b z a z y z 2 ) 
= -W + 2b • a 
Exercises: (1) Show that 

y x ay x = j£ + 2a x y x 

y^u =4 

T/i^Tju =4a-b 
y M *Wy M =-2eW 

(2) Verify by expanding in power series that 

exp [(u/2)y t y x ] = cosh (u/2) + y t y x sinh (u/2) 

exp[(0/2)y x y y ] = cos (0/2) + y x y y sin (6/2) (10-10) 

(3) Show that 

exp[-(u/2)y t y z ]y t exp [+(u/2)y t y z ] = y t cosh u + y z sinh u 



44 QUANTUM ELECTRODYNAMICS 

exp[-(u/2)y t y z ] y 2 exp [+(u/2)y t y z ] = y 2 cosh u + y t sinh u 

exp[-(u/2)y t y 2 ]y y exp[+ (u/2)y t y 2 ] = y y 

exp[-(u/2)y t y 2 ]y x exp [+(u/2)y t y 2 ] = y x (10-11) 

EQUIVALENCE TRANSFORMATION 

Suppose another representation for the y's is obtained which satisfies the 
same commutation relationships, Eq. (10-3); will the form of the Dirac equa- 
tion, Eq. (10-1), remain the same? To answer this question, make the fol- 
lowing transformation of the wave function * = S^F, where S is a constant 
matrix which is assumed to have an inverse S -1 (SS" 1 = 1). The Dirac equa- 
tion becomes 

y p 7r M S^ = mS*' (10-12) 

The 7r and S commute, since it is a differential operator plus a function 
of position, so this equation may be written 

y S7r M *' =mS*' 
Multiplying by the inverse matrix, 

S-iy Sir *' = mS~ 1 S^' 
or 

y ' 7r M ^' = m*' 

where y' = S _1 y S. The transformation y' - S _1 y S is called an equiva- 
lence transformation, and it is easily verified that the new y 's satisfy the 
commutation relationships, Eq. (10-3). Products of y 's, 

y'^v =(S- 1 y,S)(S~ 1 y y S)^S- 1 (y M y y )S 

transform in exactly the same manner as the y 's, so that equations involv- 
ing the y 's (the commutation relations specifically) are the same in the 
transform representation. This demonstrates another representation for 
the y's, and the Dirac equation is in exactly the same form as the original, 
Eq. (10-1), and is equivalent in all its results. 

RELATIVISTIC INVARIANCE 

The relativistic invariance of the Dirac equation may be demonstrated by 
assuming, for the moment, that y transforms similarly to a four-vector. 



RELATIVISTIC WAVE EQUATION 45 



That is, 

"Kx = (Tx - vy t )/(l - v 2 ) 1/2 7t = (T t " vy x )/(l - v 2 ) 1/2 

Jy = Ty 72 = Tz 

Also 7r transforms similarly to a four-vector because it is a combination of 
two four- vectors V and A . The left-hand side y tt of the Dirac equa- 
tion is the product of two four-vectors and hence invariant under Lorentz 
transformations. The right-hand side m is also invariant. Transforming 
y u as a four-vector means a new representation for the y 's, but Eqs. (10-11) 
can be used to show that the new y 's differ from the old y 's by an equiva- 
lence transformation; thus it is really not necessary to transform the y 's 
at all. That is, the same special representation can be used in all Lorentz 
coordinate systems. This leads to two possibilities in making Lorentz 
transformations: 

1. Transform the y 's similarly to a four-vector and the wave function 
remains the same (except for Lorentz transformation of coordinates). 

2. Use the standard representation in the Lorentz -transformed coordinate 
system, in which case the wave function will differ from that in (1) by an 
equivalence transformation. 



HAMILTONIAN FORM OF THE DIRAC EQUATION 

To show that the Dirac equation reduces to the Schrodinger equation for 
low velocities, it is convenient to write it in Hamiltonian form. The original 
term, Eq. (10-1), may be written 

yj-(fi/i) (3/9t) - e0]* - y • [(H/i)V-eA]* = m* 
Multiplying by cy t and rearranging terms gives 

-(fi/i)0*/at) = {y t y • [(fi/i)V - eA] + e0 + y t m}* 
= H* 
By Eq. (10-5), H is written 

H = a • [(K/i) V - eA] + ecf) + m/3 

where /3 = y t , a X) y )Z = ytVx.y.z' Ec l- ( 10_5 )» and tne a ' s satisfy the follow- 
ing commutation relations: a x 2 = a y 2 = a z 2 =/3 2 = 1 and all pairs anticommute. 
It will be noted that a,/3 are Hermitian matrices in our special represen- 
tation, so that in this representation H is Hermitian. 



46 QUANTUM ELECTRODYNAMICS 



Exercise: Show that a probability density p = &*$? and a probability 
current j = **a^ satisfy the continuity equation 

Op/at) + V- j =0 

Note: * is a four- component wave function and 

p = **# = ($f*f*f*f)[ ^ 2 ) = tff*! + *2**2 + *3** 3 + *** 
*4 






Eleventh Lecture 

It should be noted that p and a are Hermitian only in certain representa- 
tions. In particular, they are Hermitian in the representation employed thus 
far; this will be called the standard representation and expressions in it will 
be labeled S.R. when appropriate. The Hermitian property of a and £ is 
necessary in order to get 

p = \j/*\£ 

j =**a* S.R. (11-1) 

as the expressions for charge and current density. Hence they are not true 
in all representations. The Dirac equation is (with R, c restored) 

-(K/i)0¥/8t) = H* H =/3mc 2 + e0 + ca • [(fi/i) V - (e/c)A] 

(ll-2)t 



t It is noted that the Hamiltonian found in Schiff (' 'Quantum Mechanics," 
McGraw-Hill, New York, 1949) differs from this one by negative signs on all 
but the e<p term. Also the components * lf \£ 2 , ^3, ^4 of the wave function 
used in Schiff correspond, respectively, to -* b , -^ b , -* a , * a here. All 

this is the result of an equivalence transformation S 2 = i^a x cx y c^ z between 
the representations used here and in Schiff. It is easily verified that S 2 = -1 
hence S _1 = -S and 



s-i 









1 














1 


1 














-1 









RELATIVISTIC WAVE EQUATION 47 

The expected value of x is 

<x> = J **x^ d vol 

= /Ofrfx*! + **X* 2 + *3*X* 3 + * 4 *X* 4 ) d Vol S.R. 

remembering that # now is a four-component wave function. Similarly it 
may be verified as an exercise that 

<oj> = J y?*aV d vol 

< a x> = J(*4**i + ^3**2 t ***3 + ^1**4) d vol S.R. 

Also matrix elements are formally the same as before. For example, 

<«)mn =/*J«*n d Vo1 

If A is any operator then its time derivative is 

A =i(HA -AH) + 8 A/at 

For X the result is clearly 

x = i (Hx -xH) = a (11-3) 

since x commutes with all terms in H except p-oj. But a 2 = 1, so the 
eigenvalues of a are ±1. Hence the eigenvelocities of x are ± speed of 
light. This result is sometimes made plausible by the argument that a pre- 
cise determination of velocity implies precise determinations of position at 
two times. Then, by the uncertainty principle, the momentum is completely 
uncertain and all values are equally likely. With the relativistic relation be- 
tween velocity and momentum, this is seen to imply that velocities near the 
speed of light are more probable, so that in the limit the expected value of 
the velocity is the speed of light. t 
Similarly, 



(p - eA) x =i(Hp x - p x H) - ie(HA x - A^) - e9A x /at 

= -e(8 0/8x) + ea*(3A/8x)-e(a-V)A x -e(8A x /at) 
The terms in A and A x , except the last, expand as follows: 



f This argument is not completely acceptable, for X commutes with p; 
that is, one should be able to measure the two quantities simultaneously. 



48 , QUANTUM ELECTRODYNAMICS 



dA x dA v dA z dA x d\ 

x 3x y 3x 3x x 8x y oy 



2 dz J 



This seen to be the x component of 

e a x (V x A) = e a x B 
The first and last terms form the x component of E. Therefore, 



(p - eA) =e(E + a X B) = F 

where F is the analogue of the Lorentz force. This equation is sometimes 
regarded as the analogue of Newton's equations. But, since there is no di- 
rect connection between this equation and x , it does not lead directly to 
Newton's equations in the limit of small velocities and hence is not com- 
pletely acceptable as a suitable analogue. 

The following relations may be verified as true but their meaning is not 
yet completely understood, if at all: 

(d/dt) [x + (i/2m)/3a] = 03/m)(p - eA) 

(d/dt) [t + (i/2m)/3] - (/3/m)(H - e0) 

i(d/dt)(a x a y a z ) = -2mPa x a y a z 

-(d/dt)(j8a) =2((3a x a y a z )(v- eA) 

where in the last relation <r means the matrix 

a 
,0 <7 

so that 

cr z = -ia x a y , etc. 

From analogy to classical physics, one might expect that the angular mo- 
mentum operator is now 

L = Rx (p - eA) 

Note that in classical physics 

p - eA = mv (1 - v 2 r 1/2 



From previous results for R and (p - eA), the time derivative of L may be 
written 



RELATIVISTIC WAVE EQUATION 49 



L = Rx(p-eA) +Rx(p-eA) 

= a x (p - eA) + R x F 

The last term may be interpreted as torque. For a central force F, this 
term vanishes. But then it is seen that L * because of the first term; that 
is, the angular momentum L is not conserved, even with central forces. 
But consider the time derivative of the operator <7 defined as 

<J 
or 

where cr z = -a x a y , etc. The z component is seen to commute with the p, 
e<p, and a 2 terms of H but not with the a x and a y terms, so that cr z = 
+ l(Ha x a y - a x a y H) = + (a x ir x a x a y - a x a y a x n x + a y n y a x a y - OL x a y ci y -n y ) t 
where 

7r = (-iV - eA) 

But 

a x n x a x a y = a x oi x a y Tr x = oyr x 

-oi x a y a x 7r x = a x a x a y ir x = a y it x 

a y ir y a x a y = -a y a y a x ir y = -a x n y 

-a x a y a y ir y = -a x n y 

so that 

cr z - (2a y ?r x - 2a x ir y ) 

This is seen to be the z component of -2a x 7r. Finally then, 

1/2 (<r) = - a x ir = -a x(p- eA) 

and this is the first term of L with negative sign. Therefore it follows that 

(d/dt)[L+ (H/2)cr] = Rx F 

which vanishes with central forces. The operator L + (R/2)o" may be re- 
garded as the total angular momentum operator, where L represents orbi- 
tal angular momentum and (h/2)<7 intrinsic angular momentum for spin 1/2. 
Thus total angular momentum is conserved with central forces. 



50 QUANTUM ELECTRODYNAMICS 

Problems: (1) In a stationary field 0=0, 3A/at = 0, show that 

a • (p - eA) 

is a constant of the motion. Note that this is a consequence of the 
anomalous gyromagnetic ratio of the electron. It also means that the 
cyclotron frequency of the electron equals its rate of precession in a 
magnetic field. 

(2) In a stationary magnetic field 0=0, 3A/8t = 0, and for a sta- 
tionary state, show that ^ lf ^ 2 in 




are the same as ty t , ^ 2 in the Pauli equation. Also, if E Pauli is the 
kinetic energy in the Pauli equation and E Dirac - W + m is the rest 
plus kinetic energy in the Dirac equation, show that 

E Dirac = ^ 2 ™ E Pauli + ^ 

and explain the simplicity of this relationship. 



NONRELATIVISTIC APPROXIMATION TO THE DIRAC EQUATION 

It will be assumed that all potentials are stationary and stationary states 
will be considered. This makes the work simpler but is not necessary. In 
this case 

* =e~ iEt *(X) 

H* = E^ (Dirac Hamiltonian) 



and put 



That is, 



E =m + W 



H* = (m + W)* = a - (p - eA) ^ + p m* + e0^ 

It will be recalled with ^ written as Eq. (9-5) and with a, ft as given in 
Lecture 10, the previous equation may be written as two equations (9-4'), 

(m + W)* a = a- 7r* b + m* a + V* a (11-4) 



RELATIVISTIC WAVE EQUATION 51 



(m + W)* b = o- • Trt? a - m* b + V* b (11-5) 

where, as before, tt = (p - eA) and V = e0. Simplifying and solving Eq. 
(11-5) for * b gives 

* b = [l/(2m + W - V)] (a . 7r)* a (11-6) 

It is noted that if W and V are « 2m, then # b ~ (v/c)* a . For this reason 
^ a and ^ b are sometimes referred to as the large and small components of 
^, respectively. Substitution of * b from Eq. (11-6) into Eq. (11-4) gives 

W* a = (a-7r)[l/(2m+W-V)](<r.7r)* a + V* a (11-7) 

and, if W and V are neglected in comparison to 2m, the result is 

W* a = (l/2m)(<r-7r) 2 * a + V* a 

This is the Pauli equation, Eq. (9-4). 

Now the approximation will be carried out to second order, that is, to 
order v 2 /c 2 , to determine just what error may be expected from use of the 
Pauli equation. 



Twelfth Lecture 

Using the results of Lecture 11, given by Eqs. (11-6) and (11-7), the low- 
energy approximation (w - V) « 2m will be made, keeping terms to order 
v 4 . Thus 

(2m + W - V)" 1 « l/2m - (w - V)/(2m) 2 (12-1) 

Then Eq. (11-7) becomes 

(W- V)* a = (l/2m)(cr-ir) 2 * a - (l/4m 2 )(or- ir)(W- V)(<r-ir)¥ a 

(12-2) 

while the normalizing requirement J(* a 2 + * b 2 ) d vol = 1, becomes 

/* a *[l + (a-ir) 2 /(4m 2 )]* a d vol = 1 (12-3) 

By use of the substitution 

X = [1 + (cr-7T)/(8m 2 )]* a (12-4) 

the normalizing integral can be simplified to read (to order v 2 /c 2 ) 



52 QUANTUM ELECTRODYNAMICS 



J x *X d vol = 1 



This substitution also allows easier interpretation of Eq. (12-2). Rewriting 
Eq. (12-2), 

[1 + ((7-7r) 2 /(8m 2 )](W-V)[l + (cr-7r) 2 /(8m 2 )]* a 

= (l/2m)(cr-7r) 2 * a + (l/8m 2 )[((7-7r) 2 (W- V) - 2(<r -ir)(W- V) 

x (0--7T) + (W- V)(a-ir) 2 ]* a 

Then applying Eq. (12-4) and dividing by 1 + (o"-ir) 2 /(8m 2 ), there results 
(W-V)x =(l/2m)((7-7r) 2 x- (l/8m 3 )(or-7r) 4 x 

+ (l/8m 2 )[((7- 7T) 2 (W- V)-2((7'7T)(W-V)((7-7r) 
+ (W-V)((7-7r) 2 ] X (12-5) 

The techniques of operator algebra may be used to convert Eq. (12-5) to 
a form more easily interpreted. In particular one should recall that 

A 2 B - 2 ABA + BA 2 = A(AB - BA) - (AB - BA)A 
Then, since ff = (p - eA), and since 

(C7-7T)(W- V) - (W-V)(cr-7r) = -i(cr • VV) 

= +i((7 -E) 
there results [with a-ir = A and (W- V) = B in the foregoing], 

i (a • it) (a • E) - i {a • E) (a • ir) - V • E + 2a • (it x E) 

(since V x E ~ 8B/9t = here), so Eq. (12-5) can be expanded as 

W X = Vx + (l/2m)(p - eA)-(p - eA) X - (e/2m)(<7 • B) x 

(1) (2) (3) 

-(l/8m 3 )(p-p) 2 X 

(4) 

+ (e 2 /8m 2 )[V-E + 2a- (p - eA) x E] x (12-6) 

(5) C<5) 

In this form the wave equation may be interpreted by considering each 
term of Eq. (12-6) separately. 

Term (1) gives the ordinary scalar potential energy as it has appeared 
before. 



RELATIVISTIC WAVE EQUATION 53 



Term (2) can be interpreted as the kinetic energy. 

Term (3), the Pauli spin effect, is just as it appears in the Pauli equa- 
tion. 

Term (4) is a relativistic correction to the kinetic energy. The correc- 
tion derives from 

E = (m 2 + p 2 ) 1/2 = m(l + p 2 /m 2 ) 1/2 

= m + p 2 /2m - p 4 /8m 3 + '" 

The last term in this expansion is equivalent to term (4). 

Terms (5) and (6) express the spin-orbit coupling. To understand this in- 
terpretation consider the part of term (6) given by <j • (p x E). In an inverse- 
square field this is proportional to a • (p x r)/r 3 . The factor p x r can be 
interpreted as the angular momentum L to get (<j- L)/r 3 , the spin-orbit cou- 
pling. This term has no effect when the electron is in a s-state (L = 0). On 
the other hand, (5) reduces to V-E = 47rZ6(r), which affects only the s-states 
(when the wave function is nonzero at r = 0). So (5) and (6) together result in 
a continuous function for spin-orbit coupling. The magnetic moment of the 
electron e/2m, appears as the coefficient of term (3), and again of terms 
(5) and (6), i.e., (e/2m)(l/4m 2 ). 

A classical argument can be made to interpret term (6). A charge mov- 
ing through an electric field with velocity v feels an effective magnetic 
field B=vxE = (l/m)(p - eA) x E, and term (6) is just the energy (e/2m) x 
(<7-B) in this field. We get a factor 2 too much this way, however. Even be- 
fore the development of the Dirac equation, Thomas showed that this simple 
classical argument is incomplete and gave the correct term (6). The situa- 
tion is different for the anomalous moments introduced by Pauli to describe 
neutrons and protons (see Problem 3 below). In Pauli 's modified equation, 
the anomalous moment does appear with the factor 2 when multiplying terms 
(5) and (6). 

Problems: (1) Apply Eq. (12-6) to the hydrogen atom and correct 
the energy levels to first order. The results should be compared to 
the exact results. f Note the difference of the wave functions at the 
origin of coordinates. This difference actually is too restricted in 
space to have any importance. Near the origin the correct solution to 
the Dirac equation is proportional to 

r[l-(Z/13 7) 2 ] 1/2 «r- 1/40 ' 000 

for the hydrogenic atoms, while the Schrodinger equation gives ^ — 
constant as r —*■ 0. 



tSchiff, "Quantum Mechanics," McGraw-Hill, New York, 1949, pp. 323ff. 



54 QUANTUM ELECTRODYNAMICS 



(2) Suppose A and depend on time. Let W = i3/8t and follow 
through the procedures of this lecture to the same order of approxi- 
mation. 

(3) Pauli's modified equation can be applied to neutrons and pro- 
tons. It is obtained by adding a term for anomalous moments to the 
Dirac equation, thus 

y M (iV M - eA M )* +M7 M y y F Ml/ ^ = m* 

Multiplying by (3, this may be written in the more familiar "Hamil- 
tonian" expression 

i(8/8t)* = H Dirac * +ju0((7-B-a ■ E)* 

Show that the same approximation which led to Eq. (12-6) will now 
produce the terms 

[V + l/2M(p - eA) 2 + (jn +e/2M)(7-B + (l/8M 3 )(p- p) 2 + 

(1/4M 2 )(2 M +e/2M)(V-E + 2a- (p- eA)X E)] * (12-7) 

for protons, and a similar expression for neutrons, but with e = 0. 

(4) Equation (12-7) can be used to interpret electron-neutron scat- 
tering in an atom. Most of the scattering of neutrons by atoms is the 
isotropic scattering from the nucleus. However, the electrons of the 
atom also scatter, and give rise to a wave which interferes with nu- 
clear scattering. For slow neutrons, this effect is experimentally ob- 
served. It is interpreted by term (5) of Eq. (12-6) [as modified in Eq. 
(12-7) with e = 0] . Since the electron charge is present outside the 
nucleus, V- E has a value different from 0. Term (5) can be used in 
a Born approximation to compute the amplitude for neutron-electron 
scattering. However, when the effect was first discovered, it was ex- 
plained by the assumption of a neutron-electron interaction given by 
the potential c<5(R), where 6 is the Dirac 6 function and R is the neu- 
tron-electron distance. 

Compute the scattering amplitude with c6(R) by the Born approxi- 
mation and compare with that given by term (5). Show that 

c =47rjx N e 2 /4M N 2 

In order to interpret c6(R) as a potential, the average potential V 
is defined as that potential which, acting over a sphere of radius e 2 /mc 2 , 
would produce the same effect. 

Using ji n = - 1.9135 eh/2M N , show that the resulting V agrees with 
experimental results within the stated accuracy, i.e., 4400 ±400 ev.f 



tL. Foldy, Phys. Rev., 87, 693 (1952) 



RELATIVISTIC WAVE EQUATION 55 

(5) Neglecting terms of order v 2 /c 2 , show that 
/* f *af(R)*f d vol 

"* /Xf*[(pf +fp)/2m + ((7/2m)X(Vf)] Xi d vol 



Solution 

of the Dirac Equation 

for a Free Particle 

Thirteenth Lecture 

It will be convenient to use the form of the Dirac equation with the y 's 
when solving for the free-particle wave functions 

y M (iV^ - eA^)* = m* 
Using the definition of Lecture 10, # ^yu&u* 

* = y u A M = y t A t - y x A x - y y A y - y z A z 

V =r M V M =y t V t -y x V x -y y V y - y z V z 
and the Dirac equation may be written 

(iJP-e#)* =m* (13-1) 

(Recall that the quantity # = y^a^ is invariant under a Lorentz transforma- 
tion.) 

It is necessary to put the probability density and current into a four- 
dimensional form. In the special representation, the probability density and 
current are given by 

p = xjr*^ j = xj,* a ^ 



56 



SOLUTION OF THE DIRAC EQUATION 57 

If the relativistic adjointj of * is defined 

*=**£ (13-2) 

in the standard representation, then the probability density and current may 
be written 

p = 5/3* j^ = 3?y M * 

To verify this, replace * by **/3 and note that /3 2 = 1 and that (Sy^ = a^. 

Exercises: (1) Show that the adjoint of * satisfies 

#(-iJP - e#) = m* (13-3) 

(2) From Eqs. (13-1) and (13-3) show that V^ j = (conservation 
of probability density). 

In general, the adjoint of an operator N is denoted by N, and N is the 
same as N except that the order of all y 's appearing in it is reversed, and 
each explicit i (not those contained in the y's) is replaced by -i. For ex- 
ample, if N = y x y y , N= y y y x = -N. If N = iy 5 = iy x y y y z y t , then N = 
-iy t y z y y y x = -iy 5 . The following property takes the place of the Hermitian 
property so useful in nonrelativistic quantum mechanics: 

(? 2 N*i)* = (*iN* 2 ) (13-4) 

For a free particle, there are no potentials, so $ = and the Dirac 
equation becomes 

i fi* = m* 

To solve this, try as a solution 

* =ue -ip-x = ue-iPyjc,, (13_ 6) 



t* is a four -component column vector, 




The adjoint * is the four-component row vector *j*, *2* ~^3* ~^4* in the 



standard representation. Multiplication by /3 changes the sign of the third 
and fourth components, in addition to changing ** from a column vector to 
a row vector. 




58 QUANTUM ELECTRODYNAMICS 



* is a four -component wave function and what is meant by this trial solution 
is that each of the four components is of this form, that is, 



e -i P .x 



Thus Uj, u 2 , u 3 , and u 4 are the components of a column vector, and u is 
called a Dirac spinor. The problem is now to determine what restrictions 
must be placed on the u's and p's in order that the trial solution satisfy 
the Dirac equation. The V^ operation on each component of * multiplies 
each component by -ip„ , so that the result of this operation on ^ produces 

v u* =v u ue" i ^ x i; =-ip M ue- i Py x y - -ip M # 

so that Eq. (13-5) becomes 

iy M (-ip M )* = y M p M tf = #* = m* (13-7) 

Thus the assumed solution will be satisfactory if $u = rau, To simplify 
writing, it will now be assumed that the particle moves in the xy plane, so 
that 

Pi = Px P2 = Py P3 = ° P4 = E 

Under these conditions, $ = y t E - T y P y "TxPx- m standard representation 

/ o\, „\ 



1 














1 














-1 














-1 



7t I 0-1 J 7 *>v \-<7 x>y 



so i> -m becomes 




(13-8) 


By components, Eq. (13-7) becomes 

(E - mju! - (p x - ip y )u 4 = (13-9a) 

(E - m)u 2 - (p x + ip y )u 3 = (13-9b) 

(Px " iPy) u 2 - (E + m)u 3 = (13-9c) 

(p x + ip y )u 1 - (E + m)u 4 = (13-9d) 



SOLUTION OF THE DIRAC EQUATION 59 

The ratio u t /u 4 can be determined from Eq. (13-9a) and also from Eq. 
(13-9d). These two values must agree in order that Eq. (13-6) be a solution. 
Thus 

ui/u 4 = (p x -ip y )/(E - m) = (E ■+ m)/(p x + ip y ) 

or 

2 2 2 ^2 (13-10) 

p x 2 + p y 2 + m 2 = E 2 

This is not a surprising condition. It states that the p v must be chosen so 
as to satisfy the relativistic equation for total energy. 

Similarly, Eqs. (13-9b) and (13 -9c) can be solved for u 2 /u 3 giving 

u 2 /u 3 = (p x + iPy)/(E - m) = (E + m)/(p x -ip y ) 

which also leads to condition (13-10). 

A more elegant way of obtaining exactly the same condition is to start 
directly with Eq. (13-7). Then, by multiplying this equation by $ gives 

$($u) = $(mu) = m($u) = m^ 
Using Eq. (10-9), 

]^=p-p = E 2 -p x 2 -p y 2 
so that the condition becomes 

E 2 - p x 2 - p y 2 = m 2 or u = 

The former is the same condition as obtained before, and the latter is a 
trivial solution (no wave function). 

Evidently there are two linearly independent solutions of the free-particle 
Dirac equation. This is so because substitution of the assumed solution, Eq. 
(13-6), into the Dirac equation gives only a condition on pairs of the u's, 
Uj, u 4 and u 2 , u 3 . It is convenient to choose the independent solutions so that 
each has two components which are zero. Thus the u's for the two solutions 
can be taken as 





and (13-11) 



where the following notation has been used: 



60 > QUANTUM ELECTRODYNAMICS 



(13-12) 





F = 


E 


+ m 




P + = 


Px 


+ iPy 




P. = 


Px 


-ip y 


These 


solutions are 


not normalized 



DEFINITION OF THE SPIN OF A MOVING ELECTRON 

What do the two linearly independent solutions mean? There must be 
some physical quantity that can still be specified, which will uniquely deter- 
mine the wave function. It is known, for example, that in the coordinate sys- 
tem in which the particle is stationary there are two possible spin orienta- 
tions. Mathematically speaking, the existence of two solutions to the eigen- 
value equation ]6u = mu implies the existence of an operator that commutes 
with $ . This operator will have to be discovered. Observe that y 5 anticom- 
mutes with $; that is, y 5 $ = -jfy 5 . Also observe that any operator )N will 
anticommute with $ if W • p = 0, because 

W = -tifl + 2W-p (10-9) 

The combination y 5 J(Vof these two anticommuting operators is an operator 
which commutes with $; that is, 

(y 5 W = -y 5 ]6W=^(y 5 W) 

The eigenvalues of the operator (iygJ^O must now be found (the i has been 
added to make eigenvalues come out real in what follows). Denoting these 
eigenvalues by s, 

(iy 5 ^V)u = su (13-13) 

To find the possible values of s, multiply Eq. (13-13) by iy^, 

(iy 5 W)(iy 5 W)u = -y 5 Wy 5 Wu = -W-Wu = iy 5 W su = s\i 
or 

-W-W = s 2 

If W»W is taken to be -1, the eigenvalues of the operator iy 5 )V are ± 1. The 
significance of the choice W • W = -1 is as follows: In the system in which 
the particle is at rest, p x = p y = p 2 =0 and p 4 = E. Then 

= p • W = p 4 W 4 or W 4 = 




SOLUTION OF THE DIRAC EQUATION 61 



Thus, W • W = -W • W = -1 or W • W = 1. This states that in the coordinate 
system in which the particle is at rest, W is an ordinary vector (it has zero 
fourth component) with unit length. 

When the particle moves in the xy plane, choose )N to be y z , so the 
operator equation for iy 5 )V becomes 

iy 5 y z u =su 

Using relationships derived in Lecture 10, this becomes, for a stationary 
particle, f 

iy5Y*u = iy x y y y t u = iy x y y ii = - - 

\0 

This choice makes fJ the a z operator, and the relationship with spin is 
clearly demonstrated. If we define u to satisfy both $u = mu and iy5^u = su, 
this completely specifies u. It represents a particle moving with momentum 
p^ and having its spin (in the coordinate system moving with the particle) 
along the W^ axis either positive (s = +1) or negative (s = -1). 

Exercise: Show that the first of the wave functions, Eq. (13-11), is 
the s = +1 solution and the second is the s = -1 solution. 

Another way of obtaining the wave function for a freely moving electron 
is to perform an equivalence transformation of the wave function as in Eq. 
(10-12). If the electron is initially at rest with its spin up or down in the z 
direction, then the spinor for an electron moving with a velocity v in the 
spatial direction k is 



u(k) =Su u - (2m) 1/2 u u = [ v j or 



[For normalization, see Eq. (13-14).] 
From Eq. (10-11), S is given by 

S = exp[(u/2)y t y k ] cosh u = 1/(1 - v 2 ) 1/2 

Now 

exp[-(u/2)y t y k ] = cosh (u/2) + y t y k sinh (u/2) 



t For a stationary particle y t u = u. 



62 ^ QUANTUM ELECTRODYNAMICS 



and 



(2m) 1/2 cosh (u/2) = [m(l - v 2 )" 1/2 + m] 1 / 2 = (E + m) 
(2m) 1/2 sinh (u/2) = (E - m) 1/2 



Therefore, 



u (k) = [<E + m) 1/2 + y t y k (E - m) 1/2 ] u 

Writing f = (E + m), a = y t y, and noting (E 2 -m 2 ) 1/2 =p k> we get 

U( k) = (l/VF)(E+m+a -p)u 

For the case that p is in the xy plane, this just gives the result, Eq. (13-11) 
with a normalization factor 1/Vf. 

Noticing that for an electron at rest y t u = u , u (k) may be written 

(l/VF)(Ey t -y.'P + m)u 
or 

u (k) = (l/VF)(rf + m)u 
It is clear that this is a solution to the free-particle Dirac equation 

(# - m)u k = (13-7) 

for 

$ + m)(^ - m) = p 2 - m 2 = p 2 = m 2 



NORMALIZATION OF THE WAVE FUNCTIONS 

In nonrelativistic quantum mechanics, a plane wave is normalized to give 
unity probability of finding the particle in a cubic centimeter, that is, \j?*\ff = l. 
An analogous normalization for the relativistic plane wave might be some- 
thing like 

\£*>j, = u * u = uy t u = 1 

However, ^*^ transforms similarly to the fourth component of a four- 
vector (it is the fourth component of four-vector current), so this normal- 
ization would not be invariant. It is possible to make a relativistically in- 
variant normalization by setting u* u equal to the fourth component of a 



SOLUTION OF THE DIRAC EQUATION 



63 



suitable four-vector. For example, E is the fourth component of the mo- 
mentum four-vector p^, so the wave function could be normalized by 

uy t u = 2E 

The constant of proportionality (2) is chosen for convenience in later for- 
mulas. Working out (uy t u) for the s = + 1 state, 




1 








°\ 


r 





1 





1 


p 








-1 

















-1/ 


\p 



x C^ 



x Cj 2 = (F 2 + p + pJCi 2 = 2E(E + m)Ci 2 



The Cj is the normalizing factor multiplying the wave functions of Eq. 
(13-11). In order that (uy t u) be equal to 2E, the normalizing factor must 
be chosen (E + m)" 1/2 = (F)" 1 ' 2 . In terms of (uu), this normalizing condi- 
tion becomes 




(uu) = F -p 



— /T?2 



(F'-p_p + ) 



2m 2 + 2mE 
E + m 



2m 



The same result is obtained for the s = -1 state. Thus the normalizing 
condition can be taken as 



(uu) = 2m (13-14) 

In a similar manner, the following can be shown to be true: 

(uy x u) = 2p x 

(uy y u) = 2p y 

(uy z u) = 

It will be convenient to have the matrix elements of all the y's between va- 
rious initial and final states, so Table 13-1 has been worked out. 



64 



QUANTUM ELECTRODYNAMICS 



TABLE 13-1. Matrix Elements for Particle Moving in the xy Plane 



Matrix N 


s = 


1 


2m 


7x 


2p x 


?y 


2p y 


Tz 





Vt 


2E 



VF^gCuaNu!) VF 1 F 2 (u 2 Nu 1 ) Vl^ ( u 2 Nu t ) VF 1 F 2 (u 2 Nu 1 ) 
(uNu) S! = +l Sj= + 1 St = -1 S! = -1 

s 2 = + 1 s 2 = -1 s 2 = -1 S, = + 1 



F 2 F 1 - Pi + P2- 

F2P1+ + P2-F! 

-iF 2 Pi++ ip 2 -F! 

-P1 + F2+P2+FJ 

F 2 F 1 + Pi + p 2 _ 



TyTz -iF 2 F,+ ip 1 + p 2+ 

TzYx F 2 F 1 + p 1+ p 2+ 

y x y y -2iE -iF 2 F 1 -ip 1 + p 2 _ 

y t y x 2i Py F 2Pl+ -p 2 _F 1 

y t y y -2i Px -iFup^-ip^Fi 

yty z -pi+Fz-pz+f, 



y5y x =yty y yz 







y5y y =y t yzy x o 
ysy z = y t y x y y -2im 
ysy t =y x y y yz o 



-iF 2 F t -ipi + p 2 + 

FgFj - pi + p 2 + 
; iF 2 F 1 +-ip 1+ p 2 _ 

iF 2 Pi + +iF lP2+ 



«H 


CO 


+ 








II 





«T 


a 




ei 


<4-H 


fafi 


O 


3 


CD 


O 
O 


3 


X 


C? 


O 


O 


a 


O 


s 


X 





a> 









a 


<4H 


S 


O 





CD 


O 


> 








d 




bD 




CD 




fe 



y5 = y x y y y z y t ° 



iF 2 Pi+ - iF!p 2 + 



Note: p 2+ = p 2x + iP2y = P2 exp (i0 2 ); P 2 - = P2 X ~ iP2 y = P2 exp (-i0 2 ); 



F 2 = E 2 + m; F, = Ej + m; p 2 = (E - m)F. 



SOLUTION OF THE DIRAC EQUATION 65 



Limiting cases: To obtain the case where 1 is a positron at rest, the 
table gives VF2(u"2 Nu i) if one P uts F A = 0, p 1+ = 1 = Pj_ in the table. For both 
at rest as positrons, the table gives (u^Nuj) with F { = F 2 = 0; p 1+ =p 2+ = 1. 



Fourteenth Lecture 

METHODS OF OBTAINING MATRIX ELEMENTS 

The matrix element of an operator M between initial state u 1 and final 
state u 2 will be denoted by 

(u^Muj) 

The matrix element is independent of the representations used if they are 
related by unitary equivalence transformations. That is, 

u'j = Su t 

u' 2 = Su 2 

M' = SMS" 1 

u" 2 = a 2 s 



so that 



a^M'u'i - u 2 SS MS^Sut = UgMu! 



where the property S = S _1 has been assumed for S. 

The straightforward method to compute the matrix elements is simply to 
write them out in matrix form and carry out the operations. In this way the 
data in Table 13-1 were obtained. 

Other methods may be used, however, sometimes simpler and sometimes 
leading to corollary information, as illustrated by the following example. By 
the normalization convention, 

flu = 2m 
Hence 

(fl^u) = 2m 2 
since $u = mu. Similarly, 



66 QUANTUM ELECTRODYNAMICS 

(fly^tfu) =m(tfy M u) 
But also note that 

(uf*y M u) = m(fly M u) 
because fl$ = $fl = mu. Adding the two expressions, one obtains 

(fl(y M tf+l%)u) =2m(u> M u) 
From the relation proved in the exercises that 

= -W + 2a * b 

it is seen that 

*% + y u t =2p u y p =x 

But p is just a number, so it follows that 

2p p (flu) = 2m(fly p u) 
and since flu = 2m, by normalization 

(tfy M u)=2p M 

Furthermore, the general relation 

(fly t u)/(flu) = p 4 /m = E/m 
is obtained. From this it is seen why the possible normalization 

(fly^u) * E/m 

was equivalent to (flu) = 1. 

Problem: Using methods analogous to the one just demonstrated, 
show that 

(fly 5 u) =0 

INTERPRETATIONS OF NEGATIVE ENERGY STATES 

It was found that a necessary condition for solution of the Dirac equation 
to exist is 

E 2 = p 2 + m 2 



SOLUTION OF THE DIRAC EQUATION 67 



E =±(p 2 + m 2 ) 1/2 

The meaning of the positive energy is clear but that of the negative is not. 
It was at one time suggested by Schrodinger that it should be arbitrarily ex- 
cluded as having no meaning. But it was found that there are two fundamen- 
tal objections to the exclusion of negative energy states. The first is physi- 
cal, theoretically physical, that is. For the Dirac equation yields the result 
that starting with a system in a positive energy state there is a probability 
of induced transitions into negative energy states. Hence if they were ex- 
cluded this would be a contradiction. The second objection is mathematical. 
That is, excluding the negative energy states leads to an incomplete set of 
wave functions. It is not possible to represent an arbitrary function as an 
expansion in functions of an incomplete set. This situation led Schrodinger 
into insurmountable difficulties. 

Problem: Suppose that for t <0 a particle is in a positive en- 
ergy state moving in the x direction with spinup in the z direction 
(s = + 1). Then at t = 0, a constant potential A = A z (A x = A y = 0) is 
turned on and at t = T it is turned off. Find the probability that the 
particle is in a negative energy state at t = T. 

Answer: 

Probability of being in | 

negative energy state > = A 2 /(A 2 + m 2 ) sin 2 [(m 2 + A 2 ) 1/2 T] 

at t = T J 

Note that when E = -m, 1/VF" = °° , so the u's apparently blow up. 
But actually the components of u also vanish when E = — m, so that 
a limiting process is involved. It may be avoided and the correct 
results obtained simply by omitting 1/VF and replacing F by zero 
and p ± by 1 in the components of u. 

The positive energy levels form a continuum extending from E = m to +°°, 
and the negative energies if accepted as such form another continuum from 
E = — m to -°° . Between +m and — m there are no available energy levels 
(see Fig. 14-1). Dirac proposed the idea that all the negative energy levels 
are normally filled. Explanations for the apparent obscurity of such a sea of 
electrons in negative energy states, if it exists, usually contain a psycho- 
logical aspect and are not very satisfactory. But, nevertheless, if such a 
situation is assumed to exist, some of the important consequences are these: 

1. Electrons in positive energy states will not normally be observed to 
make transitions into negative energy states because these states are not 
available; they are already full. 

2. With the sea of electrons in negative energy levels unobservable, a 
"hole" in it produced by a transition of one of its electrons into a positive 
energy state should manifest itself. The manifestation of the hole is re- 
garded as a positron and behaves like an electron with a positive charge. 



68 



QUANTUM ELECTRODYNAMICS 



+ 00 

A 



+ m 



positive energy 
levels 



-m 



negative energy 
levels normally 
filled 



FIG. 14-1 

3. The Pauli exclusion principle is implied in order that the negative sea 
may be full. That is, if any number rather than just one electron could oc- 
cupy a given state, it would be impossible to fill all the negative energy 
states. It is in this way that the Dirac theory is sometimes considered as 
''proof" of the exclusion principle. 

Another interpretation of negative energy states has been proposed by 
the present author. The fundamental idea is that the "negative energy" 
states represent the states of electrons moving backward in time. 

In the classical equation of motion 

m(d 2 z M /ds 2 ) =e(dz y /ds)F My 

reversing the direction of proper time s amounts to the same as reversing 
the sign of the charge so that the electron moving backward in time would 
look like a positron moving forward in time. 

In elementary quantum mechanics, the total amplitude for an electron to 
go from x^tj to x 2 ,t 2 was computed by summing the amplitudes over all 
possible trajectories between K\,t\ and x 2 ,t 2 , assuming that the trajec- 
tories always moved forward in time. These trajectories might appear in 
one dimension as shown in Fig. 14-2. But with the new point of view, a pos- 
sible trajectory might be as shown in Fig. 14-3. 

Imagining oneself an observer moving along in time in the ordinary way, 
being conscious only of the present and past, the sequence of events would 
appear as follows: 



SOLUTION OF THE DIRAC EQUATION 



69 




FIG. 14-2 




FIG. 14-3 



1— *P 



p-t- 



— *t< 



only the initial electron present 

the initial electron still present but somewhere else an 

electron-positron pair is formed 

the initial electron and newly arrived electron and positron 

are present 

the positron meets with the initial electron, both of them 

annihilating, leaving only the previously created electron 

only one electron present 



To handle this idea quantum mechanically two rules must be followed: 



70 QUANTUM ELECTRODYNAMICS 



1. In calculating matrix elements for positrons, the positions of the ini- 
tial and final wave functions must be reversed. That is, for an electron mov- 
ing forward in time from a past state * past to a future state ^ fut , the ma- 
trix element is 



/ $ ( . Mtf „ d vol 

J fut oast 



fut past 

But moving backward in time, the electron proceeds from ¥ firf to * nast so 
the matrix element for a positron is 



i\ast M *fut dvo1 



2. If the energy E is positive, then e~ ip ' x is the wave function of an elec- 
tron with energy p 4 = E. If E is negative, e~ ip ' x is the wave function of a 
positron with energy -E or |E|, and of four -momentum -p. 



Potential Problems 
in Quantum 
Electrodynamics 

Fifteenth Lecture 

PAIR CREATION AND ANNIHILATION 

Two possible paths of an electron being scattered between the states ^ t 
and ^2 were discussed in the last lecture. These are: 

Case I. Both * lf * 2 states of positive energy, interpreted as ^i electron 
in ''past," #2 electron in "future." This is electron scattering. 

Case II. Both ¥ lf * 2 states of negative energy interpreted as ^ posi- 
tron in "future," \£ 2 positron in "past." This is positron scattering. 

The existence of negative energy states makes two more types of paths 
possible. These are: 

Case III. The ^ positive energy, * 2 negative energy, interpreted as 4^ 
in "past," ^2 positron in "past." Both states are in the past, and nothing in 
the future. This represents pair annihilation. 

Case IV. The #j negative energy, * 2 positive energy, interpreted as \frj 
positron in "future," * 2 electron in "future." This is pair creation. 





Case IV 



71 



72 QUANTUM ELECTRODYNAMICS 



The four cases can be diagrammed as shown in Fig. 15-1. Note that in 
each diagram the arrows point from % to ^ although time is increasing 
upward in all cases. The arrows give the direction of motion of the elec- 
tron in the present interpretation of negative energy states. In common lan- 
guage, the arrows point toward positive or negative time according to 
whether $ is positive or negative, that is, whether the state represented is 
that of an electron or a positron. 

CONSERVATION OF ENERGY 

Energy relations for the scattering in case I have been established in 
previous lectures. It can be seen that identical results hold for case II. To 
show this, recall that in case I, if the electron goes from the energy E t to 
E 2 and if the perturbation potential is taken proportional to exp(— icot), then 
this perturbation brings in a positive energy go. To see this, note that the 
amplitude for scattering is proportional to 

/exp(-iE 2 t)* exp(-io;t) exp(-iE 1 t) dt 

= /exp[(iE 2 t - iwt - iE t t) dt] (15-1) 

As has been shown, there is a resonance between E 2 and Ej -fa;, so that 
the only contributing energies are those for which E 2 « E± + oj. In case II 
the same integral holds but E 2 and E t are negative. A positron goes from 
an energy (past) of E past = -E 2 to an energy (future) of E fut = -E t . With the 
same perturbation energy, the amplitude is large again only if E 2 = Ej + co 
or -E past = -E fut + u), so that E fut = co + E past ; that is, the perturbation car 
ries in a positive energy co, just as it does for the electron case. 

THE PROPAGATION KERNEL 

In the nonrelativistic case (Schrodinger equation), the wave equation, in- 
cluding a perturbation potential, is written 

i9*/at = H *+ V* (15-2) 

where V is the perturbation potential and H is the unperturbed Hamilto- 
nian. For the free particle, the kernel giving the amplitude to go from point 
1 to point 2 in space and time can be shown to be 

K (2,l) =Nexp[(l/2)im(x 2 -x 1 ) 2 /(t 2 -t 1 )] t 2 >t t 

= t 2 < t A (15-3) 

where N is a normalizing factor depending on the time interval t 2 - tj and 
the mass of the particle: 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 73 

N = [m/2iri(t 2 - t t )] 1/2 

Note that the kernel is defined to be for t 2 < t t . It can be shown that K 
satisfies the equation 

[18/8 1 2 - H (2)] K (2,l) = 16(2,1) (15-4) 

The propagation kernel K v (2,l) giving a similar amplitude, but in the 
presence of the perturbation potential V, must satisfy the equation 

[18/8 1 2 - H (2) - V(2)] K v (2,l) = 16(2,1) (15-5) 

It can be shown that Ky can be computed from the series 

K V (2,D = K (2,l) - i /Ko(2,3)V(3)Ko(3,l)d 3 x 3 dt 3 

- /K (2,4)V(4)K (4,3)V(3)K (3,l)d 3 x 4 dt 4 d 3 x 3 dt 3 + •■ 

(15-6) 

In case the complete Hamiltonian H = H + V is independent of time, and 
all the stationary states n of the system are known, then K v (2,l) may be 
obtained from the sum 

K v (2,l) =2 exp[-iE n (t 2 - t^] n (x 2 )0 n *( Xl ) (15-7) 

n 

The extension of these ideas to the relativistic case (Dirac equation) is 
straightforward. By choosing a particular form for the Hamiltonian, the 
Dirac equation can be written 

i8*/8t = H* = a ■ (p - eA)* + e<f) * + m/S* 

Defining the propagation kernel as K A , then the kernel is the solution to the 
equation 

[i8/8t 2 - e0 2 - a • (-1V- eA 2 ) - mfi] K A (2,1) = i/36(2,l) 

(15-8) 

The matrix (3 is inserted in the last term in order that the kernel derived 
from the Hamiltonian be relativistically invariant. [Note the similarity to 
the nonrelativistic case, Eq. (15-6).] Multiplying this equation by /3, a sim- 
pler form results: 

{if 2 " e£ 2 - m)K A (2,l) = 16(2,1) (15-9) 

The equation for a free particle is obtained simply by letting A 2 = 0, then 
calling the free-particle kernel K + , 



74 QUANTUM ELECTRODYNAMICS 



(ift- m)K + (2,l) =16(2,1) (15-10) 

The notation K + replaces the K of the nonrelativistic case, and Eq. (15-10) 
replaces Eq. (15-4) as the defining equation. 

Just as K v can be expanded in the series of Eq. (15-6), so K A can be 
expanded as 

K A (2,1) = K + (2 f l) -i/K + (2,3)e#(3)K + (3,l)dT 3 

-/K + (2,3)e^(3)K + (3,4)e^(4)K + (4,l)dT 3 dr 4 + ••• 

(15-11) 

Note that the kernel is now a four-by-four matrix, so that all components 
of ^ can be determined. Since this is true, the order of the terms in Eq. 
(15-11) is important. The element of integration is actually an element of 
volume in four-space, 

dT = dxi dx 2 dx 3 dx 4 

The potential, -ie$(l) can be interpreted as the amplitude per cubic centi- 
meter per second for the particle to be scattered once at the point (1). Thus 
the interpretation of Eq. (15-11) is completely analogous to that of Eq. 
(15-6). 

Problem: Show that K A as defined by Eq. (15-11) is consistent 
with Eqs. (15-8) and (15-9). 

On the nonrelativistic case, the paths along which the particle reversed 
its motion in time are excluded. In the present case this is no longer true. 
The existence and interpretation of the negative energy eigenvalues of the 
Dirac equation allows the interpretation and inclusion of such paths. 

Taking t 4 > t 3 implies the existence of virtual pairs. The section from 
t 4 to t 3 represents the motion of a positron (see Fig. 15-2). 

In a time -stationary field, if the wave functions <p n are known for all the 
states of the system, then K + A may be defined by 

K + A (2,l) = 23 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) 

pos. energies 
t 2 >t, 

2 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) 

neg. energies 

t 2 < tj (15-12) 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 



75 




FIG. 15-2 



Another solution of Eq. (15-9) is 



K M2,1) = £ exp[-iE n (t 2 -t 1 )]0 n (x 2 )? n (x 1 ) 

pos. energies 

+■ 2 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) t 2 >tj 

neg. energies 







to < ti 



(15-13) 



Equation (15-13) has an interpretation consistent with the positron inter- 
pretation of negative energy states. Thus when the timing is "ordinary" 
(t 2 > tj), an electron is present, and only positive energy states contribute. 
When the timing is "reversed" (t 2 < tj), a positron is present, and only 
negative energy states contribute. On the other hand, Eq. (15-13) does not 
have so satisfactory an interpretation. Although the kernel K A defined by 
Eq. (15-13) is also a satisfactory mathematical solution of Eq. (15-9) (as 
shown below), the interpretation of Eq. (15-13) requires the idea of an elec- 
tron in a negative energy state. 

To show that both kernels are solutions of the same inhomogeneous equa- 
tion, note that their difference is 



£ exp(iE n tt) exp(-iE n t 2 )0 n (x 2 )0 n (x 1 ) 

neg. energies 

for all t 2 . This is, term by term, a solution of the homogeneous equation 
[i.e., Eq. (15-9) with zero right-hand side]. The possibility that two such 



76 



QUANTUM ELECTRODYNAMICS 



solutions exist results from the fact that boundary conditions have not been 
definitely fixed. We shall always use K + A . 

The kernel K + A , defined by Eq. (15-12), allows treatment of case III (pair 
annihilation) and case IV (pair creation) shown at the beginning of this lec- 
ture. In each case, the potential, -ie$(3), acts at the intersection of positron 
and electron paths. 



Sixteenth Lecture 



USE OF THE KERNEL K + (2,l) 

In the nonrelativistic theory it was possible to calculate the wave function 
at a point x 2 at time t 2 from a knowledge of the wave function at an earlier 
time t t (see Fig. 16-1) by means of the nonrelativistic kernel K (x2,t 2 ; x^tj), 

*(x 2 ,t 2 ) = /K (x 2 ,t 2 ; xi,t 1 )*(x 1 ,t 1 ) d 3 X! 

It might be expected that a relativistic generalization of this would be 

*(x 2 ,t 2 ) = /K + (x 2 ,t 2 ; x 1 ,t 1 )y t ^(x 1 ,t 1 ) d 3 Xl 



U 



ft 




• x 2 ,t 2 




tl 


1 
*» 



• Xo.t 



2> u 2 



-►x 



FIG. 16-1 



FIG. 16-2 



This turns out to be incorrect, however. It is not sufficient, in the relativis- 
tic case, to know just the wave function at an earlier time only because 
K + (2,l) is not zero for t 2 < tj. When the kernel is defined in this manner 
(Lecture 15), the wave function at x 2 ,t 2 (see Fig. 16-2) is given by 

*(x 2 ,t 2 ) = /K + (x 2 ,t 2 ; x 1 ,t 1 )y t ^(x 1 ,t 1 )d 3 x 1 

- /K + (x 2 ,t 2 ; x ll t 1 ')-yt*<x 1 .ti') Al tt< t 2 < V 



(16-1) 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 77 

The first term is the contribution from positive energy states at earlier 
times and the second term is the contribution from negative energy states 
at later times. This expression can be generalized to state that it is nec- 
essary to know ^(Xitj) on a four-dimensional surface, surrounding the 
point x 2 ,t 2 (see Fig. 16-3): 

*(x 2 ,t 2 ) = /K + (2 f 1)#(1)¥(1) d 4 Xl (16-2) 

where $ is the four-vector normal to the surface that encloses x 2 ,t 2 . 




FIG. 16-3 



TRANSITION PROBABILITY 

The amplitude to go from a state f to a state g under the action of a po- 
tential jL is given by an expression similar to that in nonrelativistic theory, 

a 2 l = //g(2)j8K + A (2,l) i Sf(l)d 3 x 1 d 3 x 2 (16-3) 

Using the expansion of K + A (2,l) in terms of K + (2,l), Eq. (15-12), and assu- 
ming that the amplitude for transition from state f to state g as a free par- 
ticle is zero (f and g orthogonal states), the first-order amplitude for 
transition (Born approximation) is 

a 21 = -i /g(2) j3/K + (2,3)e^(3)K + (3,l)/3f(l)dT 8 d 3 Xl d 3 x 2 

It is convenient to let 

f(3) = /K + (3,l)i3f(l)d 3 x 1 

g(3) = /g(2)/3K + (2,3)d 3 x 2 

These state that the particle has the free-particle wave function f just prior 
to scattering and the free-particle wave function g just after scattering, and 
that it eliminates any computation of the motion as a free particle. The am- 
plitude for transition, to first order, may be written 



78 QUANTUM ELECTRODYNAMICS 



-i/g(3)e£(3)f(3)dT 3 (16-4) 

(dT 3 signifies integration over time as well as space). The second-order 
term would be written 

-(1/2) //g(4)M(4)K + (4 f 3)e£(3)f<3) dr 3 dr 4 

If f(3) is a negative energy state, then it represents a positron of the future 
instead of an electron of the past and the process described by this ampli- 
tude is that of pair production. 



SCATTERING OF AN ELECTRON FROM A COULOMB POTENTIAL 

We shall make use of the theory just presented to calculate the scattering 
of an electron from an infinitely heavy nucleus of charge Ze. Suppose the 
incident electron has momentum in the x direction and the scattered elec- 
tron has momentum in the xy plane (see Fig. 16-4): 

A = 7t E i "TxPix 

jri 2 = y t E z -y x p 2x -y y p 2y 




FIG. 16-4 

The potential is that of a stationary charge Ze, 

= Ze/r, A = # = y t (Ze/r) 

The initial and final wave functions are plane waves: 

f(l) = u 1 e~ ip r x g(2) =u 2 e~ ip 2* x (four -component wave 

function) 

Thus, by Eq. (16-4), the first-order amplitude for transition from state f to 
state g (momentum p t to momentum p 2 ) is 



M = -i/u 2 e lp 2-* (Ze 2 /r)y t u 'l e "' ,pi " x d3x dt 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 



79 



Separating space and time dependence in the wave functions, this becomes 



M = -i(u 2 y t u 1 ) 



/e- i P2' x (Ze 2 /r)e i Pr x d 3 x / e^ e' iE ^ dt 



The first integral is just V(Q), a three-dimensional Fourier transform of 
the potential, which was evaluated in nonrelativistic scattering theory: 



I i(E 2 - E { ) J 



(16-5) 



V(Q) = 47rZe 2 /Q 2 Q = Pi - p 2 

The probability of transition per second is given by 

Trans, prob./sec = 27r(nN) _1 |M| 2 x (density of final states) 

(16-6) 

This is a result from time -dependent perturbation theory, the only new fac- 
tor is a normalizing factor (IIN)" 1 which takes account for the fact that the 
wave functions are not normalized to unity per unit volume. The IIN is a 
product of factors N one for each wave function, or particle in the initial 
state, and one for each final wave function, 

N = (uy t u) (16-7) 

for each particle in question. In our normalization, then N - 2E. 

The reason for this factor is that wave functions are normalized to 



(uu) = 2m 



or (uy t u) = 2E 



where, as in the computation of transition probability, they should be nor- 
malized in the conventional nonrelativistic manner ty*^ = 1 or (uy t u) = 1 
(so N = 1 for that case). 

The matrix element M, as calculated in this manner, is relativistically 
invariant and in the future the chief interest will be in M. The transition 
probability, knowing M, can be computed from Eq. (16-6). 

Density of States, Cross Section. For the electron scattering problem 
under consideration, 

M = -i(u 2 y t u 1 )(47rZe 2 /Q 2 ) 
so the transition probability is 



Trans. 



27T 



prob./sec (2E t )(2E2) 



!(u 2 y t ui)| 



47rZe' 



Q 2 



E2P2dS2 

(2tt) 3 



(16-8) 



80 QUANTUM ELECTRODYNAMICS 



where the density of final states has been obtained in the following manner: 



Density of states = 1T ^r = 4^3 
but E 2 2 = p 2 2 + ni 2 , so dp 2 /dE 2 = E 2 /p 2 and 



(27r) 3 dE 2 (27r) 3 dE 2 R " 1 



Density of states = E 2 p 2 dfi/(27r) 3 

When the incoming plane wave is normalized to one particle per cubic cen- 
timeter, the cross section is given in terms of the transition probability per 
secondt as 

Trans. prob./ sec =crv 1 == o^pj/Ej) 

or 

a = (Ej/Pi) x (trans. prob./ sec ) 

The essential difference between the relativistic treatment of scattering 
and the nonrelativistic treatment is contained in the matrix element (u^Uj). 
From Table 13-1, for a particle moving in the xy plane and s 1 = + 1, s 2 = +1, 

l(u 2 7 t ui)| 2 = 1/F 1 F 2 |F 2 F 1 + p 1 + p 2 _| 2 

where 

F 1 = F 2 = E + m 

[Ej = E 2 , conservation of energy, follows from the nature of the time integral 
in Eq. (16-5)] , and 

Pi+ =P 
P2- = Pe" i0 

(magnitude of final momentum equal to magnitude of initial momentum fol- 
lows from E} = E 2 ). 

Thus 



mvj m 2 v/ 

t Pi = n v 2x1/2 ~* Pi 2 = i v 2 — Pi 2 = ( m2 + Pl 2 ) v i 2 = E l V 

(1 — Vj ) 1 - Vj 

Therefore, v A = Pi/Ej. 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 81 

l(u 2 7tUi)| 2 = (E + m)- 2 |(E+m) 2 + p 2 e - ie | 2 

= (E ' + m) -2 {4E 2 (E +m) 2 [l - (p 2 /E 2 ) sin 2 (6/2)]} 

= (2E) 2 [1 - v 2 sin 2 (6/2)] 

When s 4 = +1, s 2 = -1 or Sj = -1, s 2 - + 1, the matrix element of y t is 
zero. When s 4 = -1, s 2 = -1, the absolute value of the matrix element is the 
same as for s t = +1, s 2 = +1. Thus spin does not change in scattering (in 
Born approximation) and the cross section is independent of spin, 

a = (4Z 2 e 4 E 2 /Q 4 )d<3 [1 - v 2 sin 2 (6/2)] Q = 2p sin (6/2) 

The criterion for validity of the Born approximation, used in obtaining this 
result, is Ze 2 /hv«l. In the extreme relativistic limit v~c. This becomes 
Z « 137. Just as for the nonrelativistic case, the scattering can actually be 
calculated exactly (correct to all orders in the potential) for the Coulomb 
potential. This exact solution of the Dirac equation involves hype rgeome trie 
functions. It was first worked out by Mott and is called Mott scattering. For 
moderate energies (200 kev) there is some probability for change in spin. 
Polarized electrons could be produced in this manner. 

Problems: (1) Calculate the Rutherford scattering law for the 
Klein-Gordon equation (particle with no spin). Result: same formula 
as just given with 1 - v 2 sin 2 (6/2) replaced by 1. 

(2) Show that this scattering formula is also correct for positrons 
(use positron states in calculating matrix element). 



Seventeenth Lecture 

CALCULATION OF THE PROPAGATION KERNEL FOR A FREE 
PARTICLE 

As shown in a previous lecture, the propagation kernel, when there is no 
perturbing potential and the Hamiltonian of the system is constant in time, 
is 

K + (2,l) =E^n(x 2 )0 n (Xi) exp[-iE n (t 2 - ti)] t 2 > tj 

+ n 

= -S^n(x 2 )?„(x 1 )exp[-iE n (t 2 - ti)] t 2 < tj 



For a free particle, the eigenf unctions <p n are 
u p exp (ip-x) 



82 QUANTUM ELECTRODYNAMICS 

and the sum over n becomes an integral over p. The u p is the spinor cor- 
responding to momentum p, positive or negative energy and spin up or down, 
as appropriate. Then the propagation kernel for a free particle is, for t 2 > tj^, 

d 3 p 1 

(2tt) 3 2E_ 

spins v ' P 



K + (2,l) = £ / 7^f £jf UpUpexptip^Xg-Xi)] 



xexp[-iE p (t 2 - t t )] 

for E p = + (p 2 + m 2 ) 1/2 . The factor l/(27r) 3 is the density of states per 
unit volume of momentum space per cubic centimeter. The factor 1/2E 
arises from the normalization uu = 2m or u-y t u = 2E p used here. The 
Up. are those for positive energy. For negative energy E p = - (p 2 + m 2 ) 1/2 , 
the Up are changed accordingly and K + (2, 1) becomes, for t 2 < tj, 

K + (2,l) = - E/t^3 ^T UpU p exp[ip-(x 2 - Xl )] 

spins \ ' P 

xexp[iE p (t 2 - t t )] 

The calculation will be made first for the case of t 2 > t t . We first calcu- 
late Up Up for positive energy, and p in the xy plane and spin up. Under 
these conditions 

E +m 




/ (E+m} l/2 

Px + iPy 

1 




(E + m) 1 



Note that u p u p is the opposite order to that usually encountered so that the 
product is a matrix, not a scalar. That is, 

(E+m) 2 (E+m)(-p x +ip y ) 





(E+m)(p x +ip y ) (Px+ip y )(-p x +iPy) 

x 1/(E + m) 

by the usual rules for matrix multiplication. But 

(Px + iPy)(-Px + iPy) = -P 2 = -E 2 + ™ 2 
and the matrix becomes 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 



83 



E + m 



Px + iPy 

By the same process, the result in the spin down case is 











-Px + iPy 


















- E+ m 




(spin up) 



(E +m) 1 / 2 



u p u p = 



(spin down) 



It may be verified easily that the sum of these matrices for spin up and spin 
down is represented by 



E Tt - PxTx - PyTy 



m 



In the general case when p is in any direction, it is clear that the only 
change is an additional term -p z y z . So, in general, 

(u p u) spin up + (u p u p ) spin down = Ey t -p-y + m = ^+m 

The sign of the energy was not used in obtaining this result so it is the same 
for either sign. 

Now put t2 - tj = t and x 2 - Xj = x. For t > 0, the propagation kernel be- 
comes 

K + (2, 1) = /(E p y t - p y + m)[d 3 p/(27r) 3 ](l/2E p ) 

x exp[-i(E p t - p-x)] 

The appearance of p in the form E p = (p 2 + m 2 ) 1/2 in the time part of the 
exponential makes this a difficult integral. Note that it may also be written 
in the form 



K + (2,i)=(iy tii+ iY x ^ + ir y - 
d 3 P 



_9_ 
8z 



/ 



(2^) 3 2E r expt-i(E p t-p.x)] 



= i(i? + m) I + (t,x) 



84 



QUANTUM ELECTRODYNAMICS 



where 



I + (t,x) =-i/ 



d^p_ 



(2tiT2E. 



exp[-i(E p t 



P-x)] 



In this form only one integral instead of four need be done. It may be veri- 
fied as an exercise that for t < the result is the same except that the sign 
of t is changed, so that putting |t| in place of t in the formula for I + (t,x) 
makes it good for all t. 

This integral has been carried out with the following result: 



I + (t,x) ="(4tt)- 1 6(s 2 ) + (m/87rs)H 1 (2) (ms) 



where s = + (t 2 
delta function and Hj 
the foregoing is 



x 2 ) 1/2 for t > x, and -i(x 2 - t 2 ) 1/2 



for t < x. 5(s ) is a 



(2) 



(ms) is a Hankel function.! Another expression for 



I + (t,x) = -(1/87T 2 ) J o °° da exp {-(i/2)[(m 2 /a0 + a(t 2 -x 2 )]} 

Both of these forms are too complicated to be of much practical use. It will 
be shown shortly that a tremendous simplification results from transforma- 
tion to momentum representation. 

Note that I + (t,x) actually depends only on |x|, not on its direction. In the 
time-space diagram (Fig. 17-1) the space axis represents |x| and the diag- 
onal lines represent the surface of a light cone including the t axis, that is, 
the accessible region of t - |x| space in the ordinary sense. It can be shown 
that the asymptotic form of I + (t,x) for large s is proportional to e~ ims . 
When one's region of accessibility is limited to the inside of the light cone, 
large s implies t 2 » |x| 2 , so that the region of the asymptotic approxima- 
tion lies roughly within the dotted cone around the t axis and is 



regions of 
asymptotic 
approximation 



surface of light 
cone (here I + is 
singular) 




Hxl 



FIG. 17-1 



tSee Phys. Rev., 76, 749 (1949); included in this volume. 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 85 



I + (t,x) — e" ims w exp{-im[t - (x 2 /2t)]} * e" Imt 

The first form is seen to be essentially the same as the propagation kernel 
for a free particle used in nonrelativistic theory. If, as in the new theory, 
possible "trajectories" are not limited to regions within the light cone, an- 
other region included in this asymptotic approximation is that within the 
dotted cone along the |x| axis where large s implies |x| 2 »t 2 . Hence 

I + (t,x) — e~ ims = exp[-im(x 2 -t 2 ) 1/2 ] « e- m l x l 

It is seen that the distance along |x| in which this becomes small is roughly 
the Compton wavelength (recall that m — - mc/fi when it represents a length" 1 
as here), so that in reality not much of the t — |x| space outside the light 
cone is accessible. 

The transformation to momentum representation will now be made. This 
is facilitated by use of the integral formula 

r oo exp (-ip 4 t) 7ri 

lim J dp 4 — o — ZTi ~ ' - - — exp (-iE_ t ) 

6 ^ J -°° ^ 4 P 4 2 "E p 2 +i€ E p ^ v V\ " 

The ie term in the denominator is introduced solely to ensure passage around 
the proper side of the singularities at p 4 2 = E p 2 along the path of integration. 
Passage on the wrong side will reverse the sign in the exponential on the 
right. 

Problem: Work out the integral above by contour integration or 
otherwise. 

Using the integral relation above, I + (t,x) becomes 



t/ ^ r d 3 p exp(-ip 4 t) exp(+ip-x) 

I + (t.X) = J —< dp 4 pt 2_ 2 +u 



But E 2 = p 2 + m 2 so this is 



t / + x _ f d4 P exp[-i(p>x)] 
I+(t ' X) " J (27T) 4 p 2 - m 2 + ic 



2 _ 



where p is now a four-vector so that d 4 p = dp 4 dp t dp 2 dp 3 , and p- = 
p^Py. Hereafter the ie term will be omitted. Its effect can be included 
simply by imagining that m has an infinitesimal negative imaginary part. In 
this form the transformation to momentum representation is easily accom- 
plished as follows (we actually take Fourier transform of both space and 
time, so this is really a momentum-energy representation): 



86 QUANTUM ELECTRODYNAMICS 



i + (p) = /l + (t,x) exp[+i(p-x)]d 4 x 

r d 4 4 d 4 x exp[-i(£-p)- x] 
J (2tt) 4 £ 2 - m 2 

where the dummy variable 4 has been substituted for p in the p integral. 
But 

J"exp[-i(£ -p) • x] d 4 x = (2tt) 4 6(£ -p) 

Hence the £ integration gives the result 

i + (p) = l/(p 2 -m 2 ) 

Finally, applying the operator i(i^ + m) to I + (t,x) gives the propagation ker- 
nel (here x = x 2 - x A ) 

K + (2,l) = W + m)I + (t,x) = i fj^i (JJT + m) ^'^^ 

. f d 4 p f) + m . .. v , 

= 1 J^ P^? exp[-i(p-x)] 

recalling that ip operating on exp[-i(p-x)] is the same as multiplying by 
$. From the identity 

1 1 ^ + m _ j> + m 

i> - m $ - m $ + m p 2 -m 2 

the kernel can also be written 

K , 9 n _ * T J^p_ exp[-i(p-x)] 
K +<2,D-i J (2*) 4 jj-m 

By the same process used for I + (t,x), the transform of K + (2,l) in momen- 
tum representation is seen to be 

k(p) = /k + (2,1) exp[+i(p-x)] d 4 x = i[l/(^ -m)] 

This is the result which was sought. 

Actually this transformation could have been obtained in an elegant man- 
ner. For K(2,l) is the Green's function of {\p -m), that is, 

(i?-m)K(2,l) =16(2,1) (17-1) 

and it is known that i'p is $ in momentum representation and 6(2,1) is unity. 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 87 



Therefore the transform of this equation can be written down immedi- 
ately: 



($ - m)k(p) = i 



or 



k(p) =i/(]rf-m) (17-2) 

as before. 

The fact that Eq. (17-1) for K(2,l) has more than one solution is re- 
flected in Eq. (17-2) in the fact that ($ - mp 1 is singular if p 2 = m 2 . We 
shall have to say just how we are to handle poles arising from this source 
in integrals. The rule that selects the particular form we want is that m be 
considered as having an infinitesimal negative imaginary part. 



Eighteenth Lecture 

MOMENTUM REPRESENTATION 

Since the propagation kernel for a free particle is so simply expressed in 
momentum representation, 

k(p) = U(i - m) 

it will be convenient to convert all our equations to this representation. It is 
especially useful for problems involving free, fast, moving particles. This 
requires four-dimensional Fourier transforms. To convert the potential, 
define 

^(q) = /^(x)exp(iq-x) d 4 x (18-1) 

Then the inverse transform is 

$(x) = (1/2tt) 4 /a'(q) exp(-iq- x) d 4 q (18-2) 

The function a(q) is interpreted as the amplitude that the potential con- 
tains the momentum (q). As an example, consider the Coulomb potential, 
given by A = 0, cp = Ze/r. 

Substituting into Eq. (18-1) gives 

tf(q) =47rZe/(Q-Q)6(q 4 )y t 

Here the vector Q is the space part of the momentum. The delta func- 
tion 6(q 4 ) arises from the time dependence of $(x). 



88 QUANTUM ELECTRODYNAMICS 

Matrix Elements. An advantage of momentum representation is the sim- 
plicity of computing matrix elements. Recall that in space representation 
the first-order perturbation matrix element is given by the integral 

M = -i/i(2)e$(2)f(l)d<r 2 
For the free particle, this becomes 

M = -i J u 2 exp(ip 2 •x 2 )e^(2)u 1 expC-ipj • Xj) dT 2 (18-3) 

In momentum representation, this is simply 

M = i(u 2 e^(q)u 1 ) (18-3') 

where ^ i s defined analogously to the three-vector q, 

The second-order matrix element in space representation is given by 

- //g(2)e#(2)K + (2,l)e#(l)f(l) dr { dr 2 

Substituting for a free particle and also expressing the potential functions as 
their Fourier transforms by means of Eq. (18-2), this becomes 

-ffff ^2 exp(ip 2 -x 2 )e^(q 2 ) exp(-iq 2 -x 2 )K + (2,l)e^(qi) 

x exp (-iqi • x{} u t exp (-ipi • xf)dTj dr 2 • d 4 q 1 /(27r) 4 

x d 4 q 2 /(27r) 4 (18-4) 

If Eq. (18-2) is used for K + (2,l), this kernel can be written 

K + (2,l) = /i/(jj-m) exp[-ip.(x 2 - Xl )] d 4 p/(27r) 4 

Writing the factors that depend on T it this part of the integral is 

J exp (ip • xj) exp (— iq t • x t ) exp (-ipj • x f ) dTj 

= (27r) 4 6 4 (p- qi - Pl ) (18-5) 

where the function 6 4 (x) is to be interpreted as 6(t 1 )6(x 2 )6(y 3 )6(z 4 ). Then 
the integral over Tj is zero for all $ except $ = $i + 4i« So the integral 
over p reduces Eq. (18-4) to 



PROBLEMS IN QUANTUM ELECTRODYNAMICS 



89 



-ffff U2 ex P (iP2 ' x 2 )e^(q 2 ) exp (-ip 2 • x 2 ) exp [-i(p! + q t ) ■ x 2 ] 

xi(^i + /4i-mrW(qi)Ui dT 2 d 4 qi /(27r) 4 d 4 q 2 /(27r) 4 

Integrating over t 2 results in another 6 function [similar to Eq. (18-5)], 
which differs from zero only when 

^2 - hi = i\ + rii 

Then integrating over d 4 q 2 gives finally 

(-i 2 )i / u 2 e^(q 2 )tfi + jzfi -mr'etfq^U! d 4 qi /(27r) 4 (18-6) 

These results can be written down immediately by inspection of a diagram 
of the interaction (see Fig. 18-1). The electron enters the region at 1 with 




FIG. 18-1 

wave function Uj and moves from 1 to 3 as a free particle of momentum jz^. 
At point 3, it is scattered by a photon of momentum 4i [under the action of 
the potential -ie^(q 1 )]. Having absorbed the momentum of the photon it then 
moves from 3 to 4 as a free particle of momentum ^ + j4i by conservation 
of momentum. At point 4, it is scattered by a second photon of momentum 
4 2 [under the action of the potential -ie^(q 2 ) absorbing the additional momen- 



90 QUANTUM ELECTRODYNAMICS 



turn # 2 )] • Finally, it moves from 4 to 2 as a free particle with wave func- 
tion u 2 and momentum $ 2 = $i + A\ + Ai- It is also clear from the diagram 
that the integral need be taken over q t only, because when jij and $ 2 are 
given, fa is determined by fa = j^ 2 -^i — fa- The law of conservation of en- 
ergy requires pj 2 = m 2 , p 2 2 = m 2 ; but, since the intermediate state is a vir- 
tual state, it is not necessary that (^ + fa) 2 = m 2 . Since the operator 
V($i + fa _m ) m ay be resolved as (fa + fa + m)/[(^ 1 + j^) 2 — m 2 ], the impor- 
tance of a virtual state is inversely proportional to the degree to which the 
conservation law is violated. 

The results given in Eqs.(18-3') and (18-6) may be summarized by the 
following list of handy rulest for computing the matrix element M = (u 2 Nuj): 

1. An electron in a virtual state of momentum $ contributes the ampli- 
tude i/($ - m) to N. 

2. A potential containing the momentum q contributes the amplitude 
-iea'(q) to N. 

3. All indeterminate momenta q { are summed over d 4 q i /(27r) 4 . 
Remember, in computing the integral, the value of the integral is desired, 

with the path of integration passing the singularities in a definite manner. 
Thus replace m by m - ie in the integrand; then in the solution take the 
limit as e -* 0. 

For relativistic work, only a few terms in the perturbation series are 
necessary for computation. To assume that fast electrons (and positrons) 
interact with a potential only once (Born approximation) is often sufficiently 
accurate. 

After the matrix element is determined, the probability of transition per 
second is given by 

P = 27r/(n N)|M| 2 x (density of final states) 

where II N is the normalization factor defined in Lecture 16. 



tSee Summary of numerical factors for transition probabilities, R. P. 
Feynman, An Operator Calculus, Phys. Rev., 84, 123 (1951); included in 
this volume. 



Relativistic Treatment 

of the Interaction 

of Particles with Light 

Nineteenth Lecture 

In Lecture 2 the rules governing nonrelativistic interaction of particles 
with light were given. The rules stated what potentials were to be used in 
the calculation of transition probabilities by perturbation theory. Those po- 
tentials are also applicable to the relativistic theory if the matrix elements 
are computed as described in Lecture 18. For absorption of a photon, the 
potential used in nonrelativistic theory was 



A M =(47re 2 ) 1/2 (2o))- 1/2 e M exp(ik-x) \ K-K -0 (19-1) 



For emission of a photon, the complex conjugate of this expression is used. 
These potentials are normalized to one photon per cubic centimeter and 
hence the normalization is not invariant under Lorentz transformations. In 
a manner similar to that for the normalization of electron wave functions, 
photon potentials will, in the future, be normalized to 2a; photons per cubic 
centimeter by dropping the (2u>)~ 1/2 factor in Eq. (19-1), giving 

A^ =(47re 2 ) 1/2 e M exp(ik-x) (19-1') 

This makes any matrix element computed with these potentials invariant, 
but to obtain the correct transition probability in a given coordinate system, 
it is necessary to reinsert a factor (2a;)~ 1 for each photon in the initial and 
final states. This becomes part of the normalization factor IIN, which con- 
tains a similar factor for each electron in the initial and final states. 

91 




92 QUANTUM ELECTRODYNAMICS 



In momentum representation, the amplitude to absorb (emit) a photon of 
polarization e^ is -i(47re 2 ) e\ The polarization vector e„ is a unit vector 
perpendicular to the propagation vector. Hence e • e = -1 and e • q = 0. 



RADIATION FROM ATOMS 

The transition probability per second is 

Trans, prob./sec = 27r |H| 2 x (density of final states) 
where H is the matrix element of the relativistic Hamiltonian, 

H=a-(-iV-eA) S.R. 

between initial and final states. That is, 

<f|H|i> = (47re 2 ) 1/2 /*f*(a -eexp (ik -x)] ^ d vol (19-2) 

Problem: Show that in the nonrelativistic limit, Eq. (19-2) reduces 
to 

l/2m J tf f *[e -p exp(ik-x) +exp(ik- x)p «e +e-(axk) 

x exp(ik-x)] ^! d vol 

This is the same result as was obtained from the Pauli equation. 

SCATTERING OF GAMMA RAYS BY ATOMIC ELECTRONS 

A relativistic treatment of scattering of photons from electrons will now 
be given. As an approximation, consider the electrons to be free (energies 
at which a relativistic treatment is necessary are, generally, much greater 
than atomic binding energies). This will lead to the Klein-Nishina formula 
for the Compton -effect cross section. 



photon 2 (outgoing) 




photon 1 (incoming 



>► x 



recoil electron 
FIG. 19-1 



INTERACTION OF PARTICLES WITH LIGHT 93 

For the incoming photon take as a potential A ljU = e ifl exp(-iq 1 -x) and for 
the outgoing photon take A 2)J = e 2jJ exp (-iq 2 - x). The light is polarized per- 
pendicular to the direction of propagation (see Fig. 19-1). Thus, 

ei • qi =0 e 2 -q 2 = 

also 

Qi ' <li =c h 2 = and q 2 • q 2 = q 2 2 = (1 9 "3) 

As initial and final state electron wave functions, choose 

^ = uj exp(-ip!-x) 

* 2 = u 2 exp(-ip 2 -x) 
where u lf u 2> p t , and p 2 satisfy 

I^jUj = muj j6 2 u 2 = mu 2 

Pi-Pt^m 2 p 2 -p 2 = m 2 (19-4) 

Conservation of energy and momentum (four equations) is written 

I*l+rii =^2+^2 (19-5) 

If the coordinate system is chosen so that electron number 1 is at rest, 

rfi = my t (19-6a) 

tf 2 = E 2 y t - p 2 cos 0y x + p 2 sin 0y y (19-6b) 

^i=^i(r t ~y x ) (i9-6c) 

Ai = w 2 (y t - Tx cos e - y y sin e ) (i9-6d) 

The latter two equations follow from the fact that, for a photon, the energy 
and momentum are both equal to the frequency (in units in which c = 1). The 
momentum has been resolved into components. The incoming photon beam 
can be resolved into two types of polarization, which will be designated type 
A and type B: 



(A) rf t = y z (B) i x = y. 



Type A has the electric vector in the z direction and type B has the elec- 
tric vector in the y direction. Similarly the outgoing photon beam can be 
resolved into two types of polarization: 



94 QUANTUM ELECTRODYNAMICS 



(A') i t = y z (B') i 2 = 7 y cos - y x sin 6 

Conservation of energy of momentum dictates that either the angle of the 
recoil electron <fi or the angle at which the scattered photon comes off 
completely determines the remaining quantities. If the electron direction is 
unimportant, its momentum can be eliminated by solving Eq. (19-5) for fa 
and squaring the resulting equation: 

fa = fa + A\ ~ Ai 

p 2 2 = m 2 = (fa + fa - fa)(fa + fa - 4 2 ) 

= Pi 2 + Qi 2 + Q2 2 + 2pi ■ q t - 2 Pl • q 2 - 2q t • q 2 

= m 2 +0+0 +2mco 1 -2maJ 2 -2cL) 1 a;2 (1 - cos 8) 

where the last line was obtained from the preceding line by using Eqs. (19-3), 
(19-4), and (19-6a, c, d). This can be written 

m(u) 1 - u) 2 ) = co 1 a; 2 (l -cos 6) 

or 

(m/a; 2 ) - (m/wj) = 1 - cos 9 (19-7) 

This is the well-known formula for the Compton shift in wavelength (or fre- 
quency) . 

DIGRESSION ON THE DENSITY OF FINAL STATES 

By the method discussed in the earlier part of the course, the following 
final state densities (per unit energy interval) can be obtained. When a sys- 
tem of total energy E and total linear momentum p disintegrates into a two- 
particle final state, 

Pl 3 d Q t 

Density of states = (27r)" J E 1 E 2 — — ~ 2 — (D-l) 

E Pl ~ E i(P - Pi) 

where E t = energy of particle 1; E 2 = energy of particle 2; pj = momentum 
of particle 1; dfij = solid angle, into which particle 1 comes out; m t = mass 
of particle 1; m 2 = mass of particle 2; and Ej + E 2 = E, pi + p 2 = p. 

Another useful formula is in terms of the final energy of particle 1 and its 
azimuth t (instead of it 0j). It is 

Density of states = (27r)" 3 (E t E 2 / |p|) dEj d<f> t (D-2) 



INTERACTION OF PARTICLES WITH LIGHT 95 

Special cases: (a) When m 2 = °° (E 2 = °°, E = °°): 

Density of states = (2tt)- 3 EjpJ dfij (D-3) 

(b) In center-of-mass system p = 0: 

Density of states = (2tt)- 3 [E 1 E 2 d^/^ +E 2 )] (D-4) 

When a system disintegrates into a three-particle final state, 

P 2 3 Pi 2 dp t dfij dfi 2 
Density of states = (2*)-* E 3 E 2 ^ - _ - _ — ; ^^ 

(D-5) 

Special case: When m 3 = °°: 

Density of states = (27r)- 6 E 2 |p 2 | dfi 2Pl 2 dp t dSl t (D-6) 

The Compton effect has a two-particle final state: taking particle 1 to be 
photon 2 and particle 2 to be electron 2, from Eq. (D-l), 

w 2 3 dfi w 

Density of states = (27T)"" 5 o^E? -; : — 2 ; 7^ 

(m + <jOi)<jo 2 - w 2 (a; 1 a; 2 cos 6) 

COMPTON RADIATION 

Calculation of |M| 2 . Using the Compton relation Eq. (19-7) to eliminate 
6, this becomes 

Density of states = (27r) -3 (E 2 w 2 3 dn o ,/ma) 1 ) 

The probability of transition per second is given by 

Trans, prob./sec =<jc= (27r/2E 1 2E 2 2a; 1 2a; 2 ) |M| 2 

x (27r)- 3 (E 2 a;2 3 dft /mc^) 



or 



a = [w 2 2 dfi ( ./(27r) 2 16m 2 w 1 2 ] |M| 2 



In working out the matrix element M, there are two ways in which the scat- 
tering can happen: (R) the incoming photon is absorbed by the electron and 
then the electron emits the outgoing photon; (S) the electron emits a photon 
and subsequently absorbs the incident photon. These two processes are 
shown diagrammatically in Fig. 19-2. 



96 



QUANTUM ELECTRODYNAMICS 



In momentum representation, the matrix element M for the first proc- 
ess R is 

i[-i(47r e 2 ) 1/2 ] 2 { u 2 ^ 2 [l/O^ + fl! - m)] i x uj 

Reading from right to left the factors in the matrix element are interpreted 
as follows: (a) The initial electron enters with amplitude Uj; (b) the elec- 
tron is first scattered by a potential (i.e., absorbs a photon); (c) having re- 





FIG. 19-2 

ceived momentum ^ from the potential the electron travels as a free elec- 
tron with momentum $ t + jz(i; (d) the electron emits a photon of polarization 
^ 2 ; and (e) we now ask for the amplitude, that the electron is in a state u 2 . 

Exercise: Write down the matrix element for the second process 
S. The total matrix element is the sum of these two. Rationalize 
these matrix elements and, using the table of matrix elements 
(Table 13-1) work out |M| 2 . 



Twentieth Lecture 

For the R diagram, M was found to be 

-i47re 2 {u 2 ^ 2 [V(^i + tfi - m)]^^} = -i47r e 2 ( u 2 Ru 1 ) 
and as an exercise the matrix element for the S diagram was found to be 



INTERACTION OF PARTICLES WITH LIGHT 97 

-i47re 2 { u 2 ^i[l/(^i -ffo-m)]^^} = -i47re 2 ( u^) 

The complete matrix element is the sum of these, so that the cross section 
becomes 

cr = (e 4 /4m)(w 2 2 /wi 2 ) dfi 2 I u 2 (R + S)u 1 | 2 

The problem now is actually to compute the matrix elements for R and S. 
First R will be considered. Using the identity 

1/(1* -m) =(]6 + m)/(p 2 -m 2 ) 

the matrices may be removed from the denominator of R giving 

(#1 +ii) - m 2mco! 

The denominator is seen to be 2mo) 1 from the following relations: 
(rfl + til) 2 - m 2 = p! 2 + 2 Pl • q t + qj 2 - m 2 
Pi = m 
qi 2 = 



2 Pi *qi = 2mw 



The matrix elements for the various spin and polarization combinations can 
be calculated straightforwardly from this point. But certain preliminary 
manipulations will reduce the labor involved. Using the identity 

M =2a-b - Vi 

it is seen that 

^2^i^i = ^2(2pre 1 ) - M\$x 

But Pt has only a time component and e t only a space component so 
Pi •e 1 = 0. Recalling that ^uj = mui, it is seen that 

u 2 ^ 2 ^i^iui = -u 2 ^2^i#iui = -(u 2 ^ 2 ^ 1 u 1 )m 

and this is the matrix element of the first term of R. It is also the negative 
of the matrix element of the last term of R, so R may be replaced by the 
equivalent 

R =^ 2 ^i^i/2mo; 1 



98 QUANTUM ELECTRODYNAMICS 

By an exactly similar manipulation, the S matrix is equivalent to 

Substituting fa = u>i(y t -y x ) and fa - w 2 (y t ~ ?x cos ^ ~ T y sin 0) and trans- 
posing the 2m factor, the complete matrix may be written 

2m(R + S) = fa(y t -y % )fa+fa(y t -y x cos 9 -y y sin 9) fa 

A still more useful form is obtained by noting that e^ anticommutes with 
Qitei'Qi = °) and ii witn ^2 and that iii\ = 2e2-e! - ^i^ 2 . Thus, 

2m(R + S) = -fa 4 x (y t - y x ) - fa fa(y t - y x + y x - y x cos - y y sin 0) 

= -2(e 2 • ei)(y t - y x ) - fa fa [y x (l - cos 0) - y y sin 0] 

Using this form of the matrix, the matrix elements may be computed easily. 
For example, consider the case for polarization: fa -y t , 4i~ 7 y cos 9 ~ y x 
sin 9. This corresponds to cases (A) and (B') of Lecture 19 and will be de- 
noted by (AB'). The matrix is 

2m(R + S) = -y z (y y cos 9 - y x sin 0)[y x (l - cos 9) - y y sin 0] 

since e 2 »ei =0. Expanded this becomes 

2m(R + S) = -y z [y y y x cos 0(1 cos 0) + cos sin0 + sin 9(1 -cos 9) 

+ y x y y sin 2 9) 

= -y z (y x y y ~ yx^cos 9 + sin 9) = -y x y y y z (l - cos 9) 

- y z sin 9 

where the anticommutation of the y 's has been used. In the case of spin- 
up for the incoming particle and spin down for the outgoing particle (s t = -1), 
s 2 = -1), the matrix elements 

-2m (FiF^ 172 ( u 2 y x y y y z Ul ) = -iF 2 p 1+ - iFj p 2 + 

-2m(F 1 F 2 ) 1/2 (u 2 y z u 1 ) = +p t + F 2 -p 2 + F t - 

may be found by reference to Table 13-1. But note that in this problem p t + 
= p xl + ip yl = since particle 1 is at rest. Hence the final matrix element 
for this case, polarization (AB 7 ), spin s A = +1, s 2 = -1, is 



>* 

^ 



>t 






•^ 



^jl 



I 

o 
o 



X 

I 






+ 



Q5 


o7 




•iH 


CQ 




CO 


o 


<D> 


X 


o 


rt 


1 


1 


CO 


CD 


N 


c^ 


CO 

o 
o 


X 


1 


»f 


1 





c*- 



+ 

•Hi 

05 






rH h 



I 

CM 

a 

as 

CQ 

O 

o 



I 

CM 

P>T 

CD 

•t-i 
CQ 



+ 




CM 




a 






+ 


pC 


CM 

a 






Q5 






Ph 


CQ 




O 


CE> 


O 


fl 


1 




1 


CQ 



pC 


+ 

CM 


a? 


P*T 


CQ 

o 
o 


Q5 

a 


1 


CQ 


tH 


1 



( t n(g+H) z ^) z J I JA ul g 



CQ 




a> 




-»-> 




CTJ 




hD 




3 




•i— > 




PS 




o 




o 




X 


CD 


J2 


> 


£ 
o 
o 


O 
^2 


a 


y. . 




r-l tH 


<D 


+ 1 


xi 




+j 


II II 


a> 


1-1 CNJ 


Sh 


CQ CQ 


aS 




H iH 


U 
O 


1 1 


«M 


II II 


CD 




CQ 


i-l CM 

CQ CQ 


O 




+-> 




<4-l 


U 


o 


O 




<4H 


CQ 


CQ 


B 


PI 

CD 

£ 


hfl 
"5* 

o 


<D 


o 


X 


X 




<D 


'u 




-t-> 


'a 


£ 


£ 

o 


CD 


o 


J3 


CD 


H 


-t-> 


oi 


CD 


o 


cd 


^ 


>> 




CD 




,fl 



100 QUANTUM ELECTRODYNAMICS 



2m(F 1 F 2 ) 1/2 (u 2 (R + S)u 1 ) =-(1 - cos 9)iF i p 2 + - sin 9 P 2+ F t 

The results for the other combinations of polarization and spin are obtained 
in the same manner and will only be presented in tabular form (Table 20-1). 
They may be verified as an exercise. 

For any one of the polarization cases listed, |M| 2 is the sum of the square 
amplitudes of the matrix elements for outgoing spin states averaged over in- 
coming spin states. But this is seen to be simply the square magnitude of 
the nonzero matrix element listed under the appropriate polarization case. 
For example, in case (AA'), 

|M| 2 = |u 2 (R + S) Ul | 2 = (l/4m 2 F 1 F 2 ) ^F^ - (1 +cos 0) F lP2 _ 
- i sin 9 FiP 2 + 
By employing the relation 

p 2 _ = Pi_ + qt_ - q 2 _ = qt_ - q 2 _ = U\ - oo 2 cos 9 + ioj 2 sin 9 
and 

(m/o; 2 ) — (m/a^) = 1 - cos 9 

the square magnitudes of the matrix elements for the various cases reduce, 
after considerable amount of algebra, to the expressions given in Table 
20-2. 

TABLE 20-2 

Polarization |M| 2 

AA' [(a)! - u> 2 ) 2 /u) jo; 2 J + 4 

AB' [(ct)t -co 2 ) 2 /o; 1 ct) 2 ] 

BA X [(W! -co 2 ) 2 /o; 1 u) 2 J 

BB 7 [(ct)! -w 2 ) 2 /o; 1 a) 2 ] + 4 cos 2 9 

It is clear that all four of these formulas may be written simultaneously in 
the form 

|M| 2 = [(wt - co 2 ) 2 /o; 1 a) 2 ] + 4(e t ■ e 2 ) 2 

Note that these formulas are "not adequate for circular polarization. That is, 
if ^! were, for example, 1/V2 - (iy z + y y ), it is seen that because of the phas- 



INTERACTION OF PARTICLES WITH LIGHT 



101 



ing represented by the imaginary part of ^ 1? all the calculations must be 
carried out before squaring the matrix elements in order to get the proper 
interference. 

Finally the cross section for scattering with prescribed plane polariza- 
tion of the incoming and outgoing photons is 

a = (e 4 /4m 2 )(a; 2 2 /co 1 2 ) dftc^ [(w 2 /^i) + (^/w 2 ) ~ 2 + 4( ei • e 2 ) 2 ] 

This is the Klein-Nishina formula for polarized light. For unpolarized light 
this cross section must be averaged over all polarizations. 

It is noted that diagram cases such as Fig. 20-1 have been included in 



di 




&i 




FIG. 20-1 



FIG. 20-2 



the previous derivation as a result of the generality in the transformation of 
of K + (2,l) to momentum representation. In fact, all diagram cases have been 
included except higher-order effects to be discussed later. (They corre- 
spond to emission and reabsorption of a third photon by the electron, such as 
in Fig. 20-2.) 



Twenty -first Lecture 

Discussion of the Klein-Nishina Formula. In the "Thompson limit," 
U3\ «m. Tnen the electron picks up very little energy in recoil, and c^^u^- 
This can be seen from the relation 



mwi - mct) 2 = k>i w 2 (1 - cos 6) 
In this limit, the Klein-Nishina formula gives 
cr=(e 4 /m 2 )(e 1 -e 2 ) 2 dn aj 



(21-1) 



(21-2) 



102 QUANTUM ELECTRODYNAMICS 



which is the Rayleigh-Thompson scattering cross section. Note that w is 
still very large compared to the eigenvalues of an atom, in accordance with 
our original assumptions for Compton scattering. 

The same result is obtained by a classical picture. Under the action of 
the electric field of the photon E = E e! exp (icot), the electron is given the 
acceleration 

a = (e/m)E e t exp (iwt) 

Classically, an accelerated charge radiates to give the scattered radia- 
tion 

_ __§_ (retarded acceleration projected on plane 1 to 
s ~ R line of sight) 

The scattered radiation polarized in the direction e 2 is determined by 
the component of the acceleration in this direction. The intensity of the scat- 
tered radiation of polarization e 2 is then (times R 2 per unit solid angle and 
per unit incident intensity) 

I=(e 4 /m 2 )(e r e 2 ) 2 (21-2') 

The customary fi 's and c's may be replaced in Eq. (21-1) as follows 
(cr is an area or length squared): 

e 4 = (e 2 ) 2 = (e 2 /Rc) 2 

m 2 = (mc/K) 2 = length squared 

e 4 /m 2 = (e 2 /mc 2 ) 2 = r 2 » 8 x 10 -26 cm 2 

Averaging over Polarizations. It is often desired to have the scattering 
cross section for a beam regardless of the incoming or outgoing polariza- 
tion. This can be obtained by summing the probabilities over the polariza- 
tions of tne outgoing beam and averaging over the incoming beam. Thus, 
suppose the incoming beam has polarization of type A. The probabilities 
(or cross sections) for the two possible types of outgoing polarization, A' 
and B' can be symbolized as AA' and AB'. The total probability for scat- 
tering a photon of either polarization is AA' + AB' . Then suppose the incom- 
ing beam is equally likely to be polarized as type A or type B. The result- 
ing probability can be obtained as the sum 1/2 (probability if type A) + 
1/2 (probability if type B). This is the situation for unpolarized incoming 
beam, and gives 

a (averaged over = U/2XAA' + AB') + (1/2)(BA' + BB') 
polarizations) 



4N d« (0* + OH- sir? e) (21-3) 



INTERACTION OF PARTICLES WITH LIGHT 103 



If, on the other hand, the polarization of the outgoing beam is measured 
(still with an unpolarized incoming beam), its dependence on frequency and 
scattering angle is given by the ratio 

Probability of p ola rization type A! _ (1/2)[AA / + BA'] 
Probability of polarization type B' ~ (1/2)[AB' + BB'] 

(fcjg/gj) + (^1/^2) 
(w 2 /wi) + (w 1 /a;2)-2sin 2 9 

The forward radiation (0 = 0) remains unpolarized, but a certain degree of 
polarization will be found in light scattered through any nonzero angle. In 
the low-frequency limit (wi^u^), the polarization is complete at 9- n/2. 
Thus an unpolarized beam becomes plane -polarized when scattered through 
90°. t 

Total Scattering Cross Section. If the cross section (averaged over polar- 
izations) given in Eq. (21-3) is integrated over the solid angle 

dfi = 27r d(cos 9) = (27rm/a; 2 2 ) da> 2 

the total cross section for scattering through any angle is obtained. So, from 
Eq. (21-1), 

cos 9=1- m/u 2 + m/u>i (2 1-1') 

and the variable a> 2 goes between the limits mcoi/(2wi + m) and uj^ as 
cos 9 goes from - 1 to + 1. Equation (21-3) can be written 

da T = (e 4 /2m 2 )(27r/o) 1 2 )m d W2 (co 2 /wi + coi/co 2 - 2m/co 2 + 2m/cj 1 

+ m 2 /o)i 2 + m 2 /o) 2 2 - 2m 2 /u>ico 2 ) 

where the last five terms replace -sin 2 6 = cos 2 9 - 1 using Eq. (2 1-1'). 
Simple integrations yield? 

o- T = 7re 4 /m 2 [(m/uj 1 -2m 2 /wi 2 -2m 3 /wi 3 ) log(2w 1 /m+l) 
+ m/2u>i + 4m 2 /wi 2 -m 3 /2w 1 (2w 1 + m) 2 ] 
In the high-frequency limit (wi"* 00 ) 
o" T ~ (l/c^i) log u)\ -* 



t Cf. Walter Heitler, "Quantum Theory of Radiation," 3rd ed., Oxford, 1954; 
and B. Rossi and K. Greissen, Phys. Rev., 61, 121 (1942). 
% Cf. Heitler, op. cit., p. 53. 



104 



QUANTUM ELECTRODYNAMICS 



Thus Compton scattering is a negligible effect at high frequencies, where 
pair production becomes the important effect. 

TWO-PHOTON PAIR ANNIHILATION 

From the quantum -electrodynamical point of view, another phenomenon 
completely analogous to Compton scattering is two-photon pair annihilation. 
Two photons are necessary (in the outgoing radiation) to maintain conser- 
vation of momentum and energy when pair annihilation takes place in the 
absence of an external potential. The interaction can be diagrammed as 
shown in Fig. 21-1. This figure should be compared to that for Compton 
scattering (Lecture 20). The only differences are that the direction of pho- 
ton fa is reversed, and, since particle 2 is a positron, ^ 2 = -(momentum of 
positron). So write 

i x = (E_y t -p. y) 

ii = -(E+Yt -p + *y) 




ii zjj 




FIG. 21-1 

where the energies E_ and E + of the electron and positron are both posi- 
tive numbers. The conservation law gives 



$i = i\~ fa- fa 



(21-4) 



(just as for Compton scattering, but the direction of fa reversed), so the 
matrix element for this interaction is 

M t = -i47re 2 (u 2 e / 2 (jfj -fa- m)- 1 ^^) 
The second possibility, indistinguishable from the first by any measure- 



INTERACTION OF PARTICLES WITH LIGHT 



105 



ment, is obtained from the first by interchanging the two photons (see Fig. 
21-2); again note similarity to Compton scattering. 
Immediately, the matrix element is 

M 2 = -i 4?re 2 ( u 2 4\ (& - ii ~ m) -1 & 2 "i) 




ii - At 




FIG. 21-2 

The sum of the two matrix elements and the density of final states gives 
the cross section 

cr • (velocity of positron) = 27r/(2E_ • 2E + • 2u 1 ' 2lo 2 ) • iMj + Mgl 2 

x (density of states) 

in a system where the electron is at rest and the positron is moving. The 
density of final states is 

oj 1 o;2/(27r) 3 co* d^ 1 /(w 2 w 1 - Q2*Qi) 

Since particle 2 is a positron, $ 2 = ~i>+, so the conservation law, Eq. (21-4), 
gives 



& + tf+ = i\ + ii 



Then 



m 2 + 2(pj • p + ) + m 2 = + 2qt ■ q 2 + 



This reduces to 



2nr + 2mE + = 2w 1 oj 2 - 2Qr 



106 QUANTUM ELECTRODYNAMICS 

Taking the velocity of the positron as |p + |/E + , the cross section is 

or= (27r)o; 1 2 dn 1 /[2E^-2|p + |4(27r) 3 -m(E + + m)] x|M! + M 2 | 2 

ufdttj |Mj + M 2 [ 2 
~64tt 2 m 2 |p + | (m + E + ) 

From a comparison of the diagrams, it is clear that the matrix elements 
for pair annihilation are the same as the matrix elements for the Compton 
effect if the sign of sk\ is changed. In the cross section, this amounts to 
changing the sign of a^. Then the cross section is 

a =e 4 cV dfi 1 /[4m 2 (E + + m)|p + |][(w 2 /a; 1 )+(a; 1 /a; 2 ) + 2 

-4(e-e 2 ) 2 ] 

in analogy with the Klein-Nishina formula. 



Twenty -second Lecture 

POSITRON ANNIHILATION FROM REST 

The formula for positron-electron annihilation derived in Lecture 21 di- 
verges as the positron velocity approaches zero (a ~ 1/v; this is true for 
other cross sections when a process involves absorption of the incoming 
particle, and is the well-known 1/v law). To calculate the positron lifetime 
in an electron density p (recall that the preceding cross section was for an 
electron density of one per cubic centimeter) as v + — - 0, we use 

Trans, prob./sec = crv+p 

plus the fact that, as v + — - 0, E + — - m and u\ ~ * w 2 ~* m (when the electron 
and positron are both approximately at rest, momentum and energy can be 
conserved only with two photons of momenta equal in magnitude but opposite 
in direction). Thus 

Trans, prob./sec =crv + p= (e 4 /2m 2 )pdft (sin 2 0) (22-1) 

where 6 = angle between directions of polarization of two photons (cos 9 
= e 1 -e 2 ). The sin 2 6 dependence indicates that the two photons have their 
polarizations at right angles. To get the probability of transition per second 
for any photon direction and any polarization, it is necessary to sum over 
solid angle (jdQ,= 47r) and average over polarizations (sin 2 6 = 1/2), giving 

Total trans, prob./sec = 1/t = (7re 4 /m 2 )p 

= 7r(e 2 /mc 2 ) 2 cp = 7rr 2 cp (22-2) 



INTERACTION OF PARTICLES WITH LIGHT 107 

(factors of c and H reinserted where required), where r = classical elec- 
tron radius, and t = mean lifetime. 

Problems: (1) Obtain the preceding result directly by using matrix 
elements for an electron and positron at rest. Show that only the sin- 
glet state (spins antiparallel) can disintegrate into two photons. The 
triplet state disintegrates into three photons and has a longer lifetime 
(see the next problem). 

(2) Find the mean time required for a positron and electron to dis- 
integrate into three photons (spins must be parallel). The following 
procedure is suggested: (1) set up formula for rate of disintegration; 
(2) write M in the simplest possible form; (3) make a table of matrix 
elements (same as Table 13-1 but with fa - my t , fa ~ — rny t ); (4) find 
the matrix element of M for eight polarization cases; (5) find the rate 
of disintegration for each case; (6) sum the disintegration rate over 
polarizations; (7) obtain the photon spectrum; (8) obtain the total dis- 
integration rate by integrating over photon spectrum and angle; and 
(9) compare with Orr and Powel.t 

(3) It is known that the matrix elements should be independent of a 
gauge transformation $ - £ + & <&> where a is an arbitrary constant 
and (;{ is the momentum of a photon whose polarization is e' or e 7 '. 
Show that substituting sk f° r ^ m the matrix elements for the Comp- 
ton effect gives m - 0. 



BREMSSTRAHLUNG 

When an electron passes through the Coulomb field of a nucleus it is de- 
flected. Associated with this deflection is an acceleration which, according 
to the classical theory, results in radiation. According to quantum electro- 
dynamics, there is a certain probability that the incident electron will make 
a transition to a different electron state with a photon emitted, while in the 
field of the nucleus. Interaction with the field of the nucleus is necessary to 
satisfy conservation of energy and momentum. That is, the electron cannot 
emit a photon and make a transition to a different electron state while trav- 
eling along in a vacuum. Figure 22-1 shows the process and defines angles 
that arise later. 

The Coulomb potential of the nucleus will be considered to act only once 
(Born approximation). The validity of this approximation was discussed in 
Lecture 16. There are two (indistinguishable) orders in which the brems- 
strahlung process can occur: (a) the electron interacts with the Coulomb 
field and subsequently emits a photon, or (b) the electron first emits a pho- 
ton and then interacts with the Coulomb field. The diagrams for these proc- 



tA. Ore and J. L. Powell, Phys. Rev., 75, 1696 (1949). 



108 



QUANTUM ELECTRODYNAMICS 



esses are shown in Fig. 22-2. The interaction with the nucleus gives mo- 
mentum to the electron. Conservation of energy and momentum requires 



i\ + $ = A + 4 



or 



= ii - i\ + i 



electron 1 

► 



Coulomb field 
of the nucleus 

FIG. 22-1 



electron 2 




photon 




f(Q) 



V(Q) — > 



i\~<A=fa + 4 



(a) 




FIG. 22-2 



In Lecture 18 it was shown that the Fourier transform of. the Coulomb poten- 
tial was proportional to 6(Q/\), since the potential is independent of time. 
This means that only transitions for which Q 4 = occur, or energy must be 
conserved among the incident electron, final electron, and photon. Thus 
Ej = E 2 + w. The transition probability is given by 



Trans, prob./sec = gv^ = (27r/2E 1 2E 2 2u;) |37l| 2 xD 



INTERACTION OF PARTICLES WITH LIGHT 109 

Since the nucleus is to be considered infinitely heavy, 

D = (2tt)- 6 E 2 p 2 dfi 2 w 2 dw dft w 

Notice that there is a spectrum of photons; that is, the photon energy is not 
determined (as it was in the Compton effect, for example). Letting SHT = 

(UjMU!), 



M = (-i)<4re*)* [* ~~z^ m) + m) -~ 



p 2 - Q - m 



(22-3) 



where the first term comes from Fig. 22-2a and the second term from Fig. 
22 -2b. The explanation of the factors in the first term, for example, is, 
reading from right to left, that an electron initially in state Uj is scattered 
by the Coulomb potential acquiring an additional momentum $ , the electron 
moves as a free particle with momentum j^ + $ until it emits a photon of 
polarization e 7 . We then ask: Is the electron in state u 2 ? For the Coulomb 
potential 

>f(Q) = (47rZe 2 /Q 2 )6(Q 4 )y t = v(Q)6(Q 4 )Yt 

(see Momentum Representation, Lecture 18) in a coordinate system in which 
the nucleus does not move. [For potential other than Coulomb, use appro- 
priate v(Q), the Fourier transform of the space dependence of the potential.] 
Rationalizing the denominator of the matrix,! 



M = (-i)(47re 2 ) l/2 v(Q) 



i\ + $ + m 

[2 y% + Y t 



L" -2p 1 -Q.-Q - 



i 2 - Q + m 




2p 2 • Q - Q 2 P _ 




(22 


-4) 



The outgoing photon can be polarized in either of two directions, and the in- 
coming and outgoing electron each have two possible spin states. The vari- 
ous matrix elements can be worked out using Table 13-1 exactly as was 
done in deriving the Klein-Nishina cross section in Lecture 20. Nothing new 
is involved, so we omit the details. After (1) summing over photon polari- 
zations, (2) summing over outgoing electron spin states, and (3) averaging 
over incoming electron spin states, the following differential cross section 
is obtained: 



t(rfi +&- m)(^i + + m) = p t 2 + 2 Pl • Q + Q 2 - m 2 = 2 Pl • Q + Q 2 

= -2 Pl • Q + Q 2 

= 2pi • Q - Q 2 Q 4 = o 



110 



QUANTUM ELECTRODYNAMICS 



1 /Ze 2 \ 2 2 dw P2 . Q ' a . fl , fl ,, 
da = — -I — o- e l sin 2 d# 2 sin 6< d8 t d<2> 

27T \ Q / W Pj 



x f p 2 2 sin 2 2 (4E 1 2 -Q 2 ) + p^sin 2 ^ (4E 2 2 -Q 2 ) 
1 (E 2 - p 2 cos 2 ) 2 (Ej - pt cos 0i) 2 

2p t p 2 sin^t sin# 2 cos (p (4E t E 2 -Q 2 + 2o; 2 ) -2a; 2 (p 2 2 sin 2 fl 2 + p t 2 sin^) 



(E 2 - p 2 cos 2 )(E 1 - pj cos t ) 



(22-5) 



An approximate expression with a simple interpretation in terms of the 
Coulomb elastic scattering cross section can be obtained when the photon 
energy is small (small compared to rest mass of electron but large com- 
pared to electron binding energies). Writing the matrix (22-3) in terms of 
gf instead of , 

m = (-imW)^ [i ^^ y<Q) + m ^^ i ] 

+ m 1 



= (-i)(47re 2 ) l/2 |rf +2v yW) + ^(Q) 



■2p 



using the relationships ^ 2 = -tf 2 & + 2e • p 2 , jz^e 7 = —4$\ + 2e • p lf and neglecting 
g[ in the numerator, since it is small, this becomes 



M«(-i)(47re 2 ) 1/2 v(Q) 



•g^Tt + 2e ; p 2 y t + m^y t 
2p 2 -q 



-Vt&h + 2prey t + me / y t 



(-i)(47re 2 ) l/2 v(Q) 



2prq 

e-p t ep 2 



6(Q 4 ) 



q*Pi q-P2 



r t 6(Q 4 ) 



where use is made of the fact that the matrix element of M between states 
u 2 and u t is to be calculated and u 2 ^ 2 = u 2 m, ^Uj = muj. 
The cross section for photon emission can then be written 



dcr = 



2E!2E 2 ' ™ }l (2tt) 3 



e 2 do; • dQ CL , /p 2 • e p t ♦ e \ 2 



7TCt) 



q q 

v P2 *^ Pi ^; 



The first bracket is the probability of transition for elastic scattering (see 
Lecture 16), so the last bracket may be interpreted as the probability of 
photon emission in frequency interval da; and solid angle d£2 w if there is 
elastic scattering from momentum p t to p 2 . 



INTERACTION OF PARTICLES WITH LIGHT 111 



Problem: Calculate the amplitude for emission of two low-energy 
photons by the foregoing method. Neglect q's in the numerator but 
not in the denominator. 

Ansiver: Another factor, similar to that in the preceding equations, 
is obtained for the extra photon. 

PAIR PRODUCTION 

It is easily shown that a single photon of energy greater than 2m cannot 
create an electron positron pair without the presence of some other means 
of conserving momentum and energy. Two photons could get together and 
create a pair, but the photon density is so low that this process is extremely 
unlikely. A photon can, however, create a pair with the aid of a field, such 
as that of a nucleus, to which it can impart some momentum. As with brems- 
strahlung, there are two indistinguishable ways in which this can happen: 
(a) The incoming photon creates a pair and subsequently the electron inter- 
acts with the field of the nucleus; or (b) the photon creates a pair and the 
positron interacts with the field of the nucleus. The diagrams for these al- 
ternatives are shown in Fig. 22-3. The arrows in the diagram indicate that 




nQ) 




(a) (b) 

FIG. 22-3 

^! is the positron momentum and ^ 2 * s the electron momentum. Notice that, 
with respect to the directions that the arrows point (and without regard to 
direction of increasing time), these diagrams look exactly like those for the 
bremsstrahlung process: Starting with fa in case (a), the particle is first 
scattered by the Coulomb potential and then by the photon; in case (b) the 
order of the events is reversed. The difference between pair production and 
bremsstrahlung, when the direction of time is taken into account, is (1) fa is 
a positron state (an electron traveling backward in time), and (2) the photon 
^ is absorbed rather than emitted. As a result, the bremsstrahlung matrix 
elements can be used for this process if fa is replaced by -$ + and ^ by -&. 



112 



QUANTUM ELECTRODYNAMICS 



The ^ + is then the positron momentum and <& is the momentum of the ab- 
sorbed photon. The density of final states is different, of course, since the 
particles in the final state are now a positron and electron. Thus 

da= (l/27r)(Ze 2 /Q 2 ) 2 e 2 (p + p_ sin0 + d0 + sin0_ d0_ d<p/u 3 ) 

x{ } (22-6) 

where the braces are the same as for bremsstrahlung, Eq. (22-5), except 
for the following substitutions: 



P- for p 2 
"P+ for Pi 



-6_ for e 2 
-0 + for 6_ 



E_ for E + 
-E + for E t 
-gl> for u 



Figure 22-4 defines the angles ($ = angle between electron-photon plane and 
positron-photon plane). 

positron 



electron 




photon 



FIG. 22-4 



Twenty -third Lecture 

A METHOD OF SUMMING MATRIX ELEMENTS OVER SPIN STATES 

By using current methods of computing cross sections, one first arrives 
at a cross section for "polarized" electrons, that is, electrons with definite 
incoming and outgoing spin states. In practice it is common that the incom- 
ing beam will be "unpolarized" and the spins of the outgoing particles will 
be unobserved. In this case, one needs the cross section obtained from that 
for "polarized" electrons by summing probabilities over final spin states 
and averaging this sum over initial spin states. This is the correct process 
since the final spin states do not interfere and there is equal probability of 
initial spin in either direction. Formally, if 



(T-U^MU!) 



INTERACTION OF PARTICLES WITH LIGHT 113 



one needs 



\ Tj Tj l^ 2 Mu 1 )| 2 (23-1) 



2 

spins 1 spins 2 

where Yj me ans the sum over final spin states for only one sign of the 

spins 2 

the energy, that is, over only two of the four possible eigenstates. Similarly, 
Y) is the sum over initial spins for one sign of the energy. The purpose 

spins 1 

now is to develop a simple method for obtaining these sums . 

In accordance with the usual rule for matrix multiplication, the following 
is true: 

Yj (u 2 Au 1 )(u 1 Bu 2 ) = 2m(u 2 ABu 2 ) (23-2) 

all u x 

where A and B are any operators or matrices, the 2m factor on the right 
arises from the normalization uu = 2m, and the sum is over all eigenstates 
represented by uj. But the states u, which we want in Eq. (23-1) are not all 
states, just those satisfying i>\\\\ = mu t . That is, they belong to the eigen- 
value + m of the operator jfj. Since ^i 2 = m 2 , ^ also has the eigenvalue -m, 
that is, there are two more solutions of jz^u = -mu which, together with the 
two we wish in Eq. (23-1) bring the total to four. Let us call the latter 
" negative eigenvalue" states. 

Now, if in Eq. (23-2) the matrix elements of B were zero in negative 
eigenvalue states, this would be the same as £y » that is, just over posi- 
tive eigenvalue states. So consider spins 1 



2} (u 2 Au 1 )(u 1 (^ 1 + m)Bu 2 ) = (u 2 A(^ 1 + m)Bu 2 )2m 

all u t 



But 



u^^+m) = for negative eigenvalue states 

= Uj(2m) for positive eigenvalue states 

so the preceding sum also equals 

X) (u 2 Au 1 )2m(u 1 Bu 2 ) 

spins 1 

Cancelling the 2m factors, this gives 

2 (u 2 Au 1 )(u 1 Bu 2 ) = (u 2 A(& + m)Bu 2 ) 

spins 1 

(^! + m) is called a projection operator for obvious reasons. Similarly it 
follows that 



114 QUANTUM ELECTRODYNAMICS 

£ (u 2 Xu 2 ) = £ <l/2m)(u 2 0fe + m)Xu 2 ) 

spins 2 all u 2 

where X is again any matrix. Remembering the normalization u 2 u 2 - 2m, 
it is seen that the last sum is just the trace or spur of the matrix $ 2 + m)X. 
Note that the order of X and $ 2 + m is immaterial. 
Finally, when one wants 

2 2 |<u 2 m Ui )1 2 

spins 1 spins 2 

collection and specialization of the previous results is seen to give 

1/2 £ £ |u 2 MUi| 2 »l/2 £ £ (uaMutM^Muz) 

spins 1 spins 2 spins 1 spins 2 

= 1/2 Sp[(^ 2 + m)M(^ 1 + m)M] 

(23-3) 

where the last notation means the spur of the matrix in the brackets. It is 
true whether $j, $ 2 represent electrons or positrons. 

The following list of the spurs of several frequently encountered matrices 
may be verified easily: 

Sp[l] = 4 Sp[y M ] = Sp[xy] = Sp[yx] 

Sp[x + y] =Sp[x] + Sp[y] 

Spiral = ° if M * v 

= +4 if jit = v = 4 

= -4 if jli = v= 1, 2, 3 

SpMJfl = 1/2 Sptftf + Kd] = Sp[a • b] = 4 a • b 

SpLa^ = 

It is also true that the spur of the product of any odd number of daggered 
operators is zero. 

Sp[(^ 1 + m 1 )(^ 2 -m 2 )] = Sp[^ 2 ] + Sp[m^ 2 - ^im 2 - m^] 

= 4(p 1 -p 2 - m^j) (23-4) 

Sp[$!+ m!)(^ 2 - m 2 )(tf 3 + m 3 )$4-m 4 )] 

= 4(p 1 -p 2 -m 1 m 2 )(P3 , P4-m 3 m4) - 4(pj • p 3 - m^g) 

x (p 2 • p 4 - m 2 m 4 ) + 4(pj • p 4 - m 1 m 4 )(p 2 ■ p 3 - m 2 m 3 ) (23-5) 



INTERACTION OF PARTICLES WITH LIGHT 115 



As an example, the case of Coulomb scattering will be ' 'treated" using 
this technique. The cross section for polarized electrons was previously 
found to be 

cr=(Z 2 e 4 /Q 4 )|(u 2 y t u 1 )| 2 

Therefore, since y t = y t , the cross section for unpolarized electrons is, by 
Eq. (23-3), 

^unpoi = 1/2 (Z 2 e 4 /Q 4 ) Sp[(^ 2 + mfrtOfc + m)y t ] 

The spur can be evaluated immediately from Eq. (23-5) with m 2 = m 4 = and 
F*2 = $4 = Tt • Another way is: Since y t ^i = 2Ej - $iJ t , it is seen that 

$ 2 + m)y t (& + m)y t = $ 2 + m)(2E 1 y t - jf t + m) 

Using a few of the formulas listed previously, the spur of this matrix is 
seen to be 

- 4 Pl *P2 + 8E 1 E 2 + 4m2 

But Pi • p 2 = E^ - Pi • P2 , Pi* P2 = P 2 cos 9, and Ej = E 2 , so this is 

4E 2 + 4m 2 + 4p 2 cos 
Also m 2 = E 2 — p 2 , so that finally the cross section becomes 



= 1/2 (Z 2 e 4 /Q 4 )[8E 2 + 4p 2 (cos 9 - 1)] 



unpol 

-2«4 /^4x T72 



= (4Z^eVQ 4 )EMl - v £ sin' (6/2)] 

where v 2 = p 2 /E 2 . This is the same cross section obtained previously by 
other methods. 



EFFECTS OF SCREENING OF THE COULOMB FIELD IN ATOMS 

The cross sections for the pair production and bremsstrahlung processes 
contained the factor [V(Q)] 2 , where V(Q) is the momentum representation 
of the potential; that is, 

V(Q) = /v(R) exp (-iQ • R) d 3 R 
which for a Coulomb potential is 

V(Q)=47rZe 2 /Q 2 
where Q is the momentum transferred to the nucleus or p x - p 2 - q. 



116 QUANTUM ELECTRODYNAMICS 



Clearly V(Q) gets large as Q gets small. The minimum value of Q oc- 
curs when all three momenta are lined up (Fig. 23-1): 

Pi P2 




lQ min l = pi - p 2 - q 

= Ipil - I'PsI - ( E i - E 2) 
For very high energies E »m, 

E - p « m 2 /2E 
so that in this case 

Qmin = (m 2 /2)[(l/E 2 ) - (1/E,)] « mV2EtE 2 

From this it is seen that Q min ~* as E| -*• °° . This shows clearly why the 
cross sections for pair production and bremsstrahlung go up with energy. 

From the integral expression for V(Q) it is seen that the main contribu- 
tion to the integral comes when R ~ l/Q. So as Q becomes small the im- 
portant range of R gets large. It is in this way that screening of the Cou- 
lomb field becomes effective. The value of 1/Q min for a contemplated proc- 
ess can be estimated from the foregoing formula. The atomic radius is 
given roughly by a Z~ 1/3 , where a is the Bohr radius. Thus if 

Re.f = l/Q min > a„Z- 1/3 

or, what is the same, 

EjEg/q > 1/2 (137) mZ*" l/3 

then screening effect will be important, and vice versa for the opposite in- 
equalities. If from this estimate screening would appear to be important, 
one should use the screened Coulomb potential. It gives the result 

V(Q) = (47re 2 /Q 2 )[Z - F(Q)] 
where F(Q) is the atomic structure factor given by 

F(Q) = /n(R) exp(-iQ'R) d 3 R 
and n(R) is the electron density as a function of R. 



INTERACTION OF PARTICLES WITH LIGHT 



117 



Twenty -fourth Lecture 

Problem: In discussing bremsstrahlung it was found that the cross 
section for emission of a low-energy photon can be approximated as 



cr = oq e 2 47r dft (du;/7rc<;)[p2 ' e /P2 ' (q/w) - Pi ' e/pi (q/co)]' 



(24-1) 



where o" is the scattering cross section (neglecting emission). Now 
consider an energetic Compton scattering in which a third, weak 
photon is emitted. The three diagrams are shown in Fig. 24-1. 




(weak) 




(weak) 




(weak) 



FIG. 24-1 

Show that the cross section for this effect is given by Eq. (24-1), with 
the Klein-Nishina formula replacing cr . (Remember to assume q 
small.) 




potential 
region 




FIG. 24-2 



FIG. 24-3 



Interaction of 
Several Electrons 



Even though the Dirac equation describes the motion of one particle only, 
we can obtain the amplitude for the interaction of two or more particles 
from the principles of quantum electrodynamics (so long as nuclear forces 
are not involved). 

First consider two electrons moving through a region where a potential 
is present and assume that they do not interact with one another (see Fig. 
24-2). The amplitude for electron a moving from 1— ~3, while electron b 
moves from 2 — * 4 is given the symbol K(3,4;l,2). If it is assumed that no 
interaction between electrons takes place, then K can be written as the 
product of kernels K + (a) (3,l) K + (b) (4,2), where the superscript means that 
K + (a) operates only on those variables describing particle a, and similarly 
for K + (b) . 

A second type of interaction gives a result indistinguishable from the 
first by any measurement in accordance with the Pauli principle. This dif- 
fers from the first case by the interchange of particles between positions 3 
and 4 (see Fig. 24-3). Now the Pauli principle says that the wave function of 
a system composed of several electrons is such that the interchange of space 
variables for two particles results in a change of sign for the wave function. 
Thus the amplitude (including both possibilities) is K = K + (a) (3,l) K + (b) (4,2) - 
K + (a) (4,l) K + (b) (3,2). 

A similar situation arises in the following occurrence. Initially, one elec- 
tron moves into a region where a potential is present. The potential creates 
a pair. Finally one positron and two electrons emerge from the region. 
There are two possibilities for this occurrence, as shown in Fig. 24-4. 
Again, the total amplitude for the occurrence is the difference between the 
amplitudes for the two possibilities. 

118 



INTERACTION OF SEVERAL ELECTRONS 



119 




potential 
region 




FIG. 24-4 

The probability of this occurrence, or the previous, or any other similar 
occurrence is given by the absolute square of the amplitude times the num- 
ber P v . The P v is actually the probability that a vacuum remains a vac- 
uum; because of the possibility of pair production, it is not unity. The P v 
can be computed by making a table of the probabilities of starting with noth- 
ing and ending with various numbers of pairs, as is shown in Table 24-1. 





TABLE 24-1 


Final number 




of pairs 


Probability 





Pvl 2 


1 


P V |K + (2,1)| 2 


2 


P v |K + (3,1) K + (4,2) -K + (4,l) K + (3,2)| 2 


3 


etc. 


etc. 





The sum of all these probabilities must equal unity, and P v is determined 
from this equation. The magnitude of P v depends on the potential present. 
So the "probabilities" taken as merely the squares of amplitudes (that is, 
omitting the P v factor) are actually relative probabilities for various oc- 
currences in a given potential. 

Use of 6 + (s 2 ). For the present, the existence of more than one possibility 
for an occurrence (the Pauli principle) will be neglected. The total ampli- 
tude can always be derived from one by interchanging the proper space var- 
iables, making the corresponding changes in sign, and summing all the am- 
plitudes so obtained. 



120 QUANTUM ELECTRODYNAMICS 



The nonrelativistic Born approximation to the amplitude for an interac- 
tion is 



K(3,4; 1,2) = K (0) + K (1) 
where, from earlier lectures, 

K (0) = K (a) (3,l)K (b) (4,2) 
and 

K (1) = -i/K (0) (3,4;5,6)V(5,6)K (0) (5,6; 1,2) d 3 X 5 d 3 X 6 dt 5 

Note that t 5 = t 6 since a nonrelativistic interaction affects both particles 
simultaneously. The potential for the interaction is the Coulomb potential 

V(5,6) =e 2 /r 5)6 

Separate variables may be used for t 5 and t 6 , if the function 6(t 5 — t 6 ) is 
included as a factor. Then 

K (1) --i//Ko(3,5)K (4,6)(e 2 /r 5 ,6)5(t5-t 6 )K (5,l)K (6,2) 

x dT 5 dT 6 

where the differential dr includes both space and time variables. It is con- 
ceivable that the relativistic kernel could be obtained by substituting K + for 
K , and introducing the idea of a retarded potential by replacing 6(t 5 - t 6 ) 
by 6(t 5 - t 6 -r 56 ). However this 5 function is not quite right. Its Fourier 
transform contains both positive and negative frequencies, whereas a photon 
has only positive energy. Thus 

6(X) = f 00 ^ exp(-iwX) dcj/27r 

To correct this, define the function 

6 + (X) = /f exp(-icoX) dw/ir 

which contains only positive energy. The value, of the function is determined 
by the integral. Thus, 

6 + (X) = lim (l/7ri)(X - ie) 

e-0 

= 6(X) + (l/7ri)(principal value 1/X) 

Abbreviating t 5 - t 6 = t and r 5 6 = r, and taking account of the fact that both 
t 5 =£ t 6 and t 5 a: t 6 are possible, the retarded potential is 



INTERACTION OF SEVERAL ELECTRONS 



121 



V(5,6) = (e 2 /2r)[6 + (t-r) + 6 + (-t - r)] 

Exercises: (1) Show that 

(l/2r)[6 + (t - r) + 6 + (-t - r)] = 6 + (t 2 - r 2 ) 

Defining t 2 - r 2 as s 5 6 2 , a relativistic invariant, the potential is 
e 2 <5 + (s 5)6 2 ). Another term which must be included is the magnetic in- 
teraction, proportional to — V a • V b . In the notation used for the Dirac 
equation, this product is — a a • a h . It will be found convenient to ex- 
press this in the equivalent form -(/3oO a ■ (j3a) b , and in this notation 
the retarded Coulomb potential is proportional to /3 a j3 b . These (3 's 
come from the use of the relativistic kernel. Thus the complete po- 
tential for the interaction becomes 

e 2 6 + (s 5>6 2 )[/3 a - i S b -(/3a) a - (jS«) b ] = e 2 6(s 5>6 2 ) y/ a) r M (b) 

and then the first-order kernel is 

K< 1) (3,4;1,2) = -ie 2 //K + (a) (3,5) K + (b) (4,6)7 M (a) y/i (b) 

x 6 + (s 5>6 2 ) K + < a >(5,l) K + (b) (6,2) dr 5 dr 6 

= -ie 2 //[K + (3,5)y M K + (5,l)] a 6 + (s 5>6 2 ) 



x [K + (4,6)y u K + (6,2)] b dr 5 dr 6 



(24-2) 



Here the superscript on y M indicates on which set of variables the 
matrix operates, just as for the superscripts on K + . 

The occurrence represented by this kernel can be diagrammed as 
in Fig. 24-5. This represents the exchange of a virtual photon be- 




FIG. 24-5 



122 QUANTUM ELECTRODYNAMICS 



tween the electrons. The virtual photon can be polarized in any one of 
the four directions, t, x, y, z. Summation over these four possibilities 
is indicated by the repeated index of 7^7^. The integral expression for 
the kernel, Eq. (24-2), implies that the amplitude for a photon to go 
from 5 — 6 (or from 6 — * 5 depending on timing) is 6 + (s 56 2 ). Equation 
(24-2) can be taken as another statement of the fundamental laws of 
quantum electrodynamics. 
(2) Show that 

6 + (s 2 ) =-47rJ[exp(-ik-X)] d 4 k/(k 2 + ie)(27r) 4 

Thus, in momentum space, 

6 + (s 2 )— -4?r/k 2 



Twenty -fifth Lecture 

DERIVATION OF THE "RULES" OF QUANTUM ELECTRODYNAMICS 

From the results of the last lecture, it is evident that the laws of electro- 
dynamics could be stated as follows: (1) The amplitude to emit (or absorb) 
a photon is ey^ , and (2) the amplitude for a photon to go from 1 to 2 is 
^+( s i,2 2 )> where 

6 ( S 2>=-47r fc - ik '< x 2- x i) d4k / 25 ,v 

MHl>- 47T j e ^ - - —r (25 1) 

= -4?r/(k 2 + ie) 

in momentum representation. It is interesting to note that S + (s 1>2 2 ) is the 
same as I + (s lj2 2 ), the quantity appearing in the derivation of the propagation 
kernel of a free particle, with m, the particle mass, set equal to zero. A 
more direct connection with the Maxwell equations can be seen by writing 
the wave equation, □ 2 A fi = 47r J^ in momentum representation, 

-k 2 a M = 4ttj m or sl u = -(47r/k 2 ) j^ (25-2) 

We now consider the connection with the "rules" of quantum electrody- 
namics given in the second lecture. The amplitude for a to emit a photon 
which b absorbs will now be calculated according to those rules (see Fig. 
25-1). The amplitude that electron a goes from 1 to 5, emits a photon of 
polarization i and direction K, then goes from 5 to 3 is given by 

[K + (3,5) i V(47re 2 /2K) exp(-iK'r 5 ) exp (iKt 5 )K + (5,l)] a 



INTERACTION OF SEVERAL ELECTRONS 



123 




whereas the amplitude that b goes from 2 to 6, absorbs a photon of polari- 
zation 4 and direction K at 6, then goes from 6 to 4 is given by 



[K + (4,6) * V(47reV2K) exp (iK • r 6 ) exp (-iKt 6 ) K + (6,2)] b 



The amplitude that both these processes occur, which is equivalent to b ab- 
sorbing a's photon if t 6 > t 5 is just the product of the individual amplitudes. 
If a absorbs b's photon, the signs of all the exponentials in the preceding 
amplitudes are changed and t 6 must be less than t 5 . 

To obtain the amplitude that any photon is exchanged between a and b, it 
is necessary to integrate over photon direction, sum over possible photon 
polarizations, and integrate over t 5 and t 6 , subject to the aforementioned 
restrictions. In summing over polarizations, $ will be replaced by y„ and 
a summation over pt will be taken. This amounts to summing over four di- 
rections of polarization, something that will be explained later. Thus 

f Amp. fori 

\ photon \ = 47re 2 £ J exp [-iK ■ (r 5 - r 6 )l exp [iK(t 5 - 1 6 )] 
[a — b J " 

x [K + (3,5)y M K + (5,l)] a [K + (4,6)^K + (6,2)] b 



x (l/2K)[d 3 K/(27r) 3 ] dt 5 dt ( 



t 6 >t 5 



47re 2 2 /exp [iK ■ (p 5 - r 6 )] exp [-iK(t 5 - t 6 )j 
u 

x [K + (3,5)y^K + (5,l)]JK + (4,6)y p K + (6,2)] b 



x (l/2K)[d 3 K/(27r) 3 ] dt 5 dt 6 



t 6 < t 5 (25-3) 



124 



QUANTUM ELECTRODYNAMICS 



Comparing this with the result of the last lecture, it must be that 

<5 + (s 5 , 6 2 ) = 4tt J' exp [-iK ■ (r 5 - r 6 )] exp [iK(t 5 - 1 6 ) (1/2K) 



x [d 3 K/(27T) 3 ] t 6 >t 5 



= 4tt j exp [iK • (r 5 - r 6 )] exp [-iK(t 5 - 1 6 )] (1/2K) 



x [d 3 K/(27i) 3 ] 



t 6 <t 5 



This can be written in a form which makes the space -time symmetry evi- 
dent by using the Fourier transform 

exp(-iK|t|) = J2o [2iK/(w 2 -K 2 + ie)] exp (-iwt) dw/27r 

so that the foregoing equation becomes 



6+(s56 2 ) = ^ 47r / expt-ik-(x 5 -x 6 )] d^k 
+ 5 ' 6 ' J k 4 2 -K-K+ie (2ti) 4 



(25-4) 



and comparing this with the result of the last problem of Lecture 24 estab- 
lishes that the rules given in Lecture 2 are consistent with relativistic elec 
trodynamics developed in the last lecture. 

ELECTRON-ELECTRON SCATTERING 

The theory will now be used to obtain the electron-electron scattering 
cross section. The diagrams for the two indistinguishable processes are 
shown in Fig. 25-2. 




& 







FIG. 25-2 



INTERACTION OF SEVERAL ELECTRONS 125 



The amplitude expressed in momentum representation is obtained as 
follows: Write Eq. (25-3) [with the aid of Eq. (25-4)] as 

e 2 E/[ K + ( 3 ' 5 )^K + (54)] a ^[K + (4,6)y M K + (6,2)] b ^4 

x dT 5 dT 6 

Since electron state 1 is a plane wave of momentum ^i and electron state 3 
is a plane wave of momentum $ 3 , it is clear that in momentum representa- 
tion the spinor part of the first bracket will become (i^y^Ut) and the spinor 
part of the second bracket will become (u 4 y^u 2 ). Integration over T5 and tq 
produces the conservation laws given at the bottom of the diagrams. Drop- 
ping the integration over q puts the photon propagation in momentum repre- 
sentation directly. Thus the matrix element can be written 



M = + i47re 2 



(u 4 y ti u 2 )(u 3 y a u 1 ) (u 4 y u u 1 )(u 3 y |tx u2) 
(fa " fa) 2 (fa " fa) 2 



The first term comes from diagram R, the second from diagram S, and the 
summation over /u is implied. In the center-of-mass system, the probability 
of transition per second is 



Trans, prob./sec = crvt = 7^74 l M l /0 _ 3 OT7 ,_2 



(2E)* '"""' (2;r) 3 2Ep' 



(see Density of Final States, Lecture 19). The method of Lecture 23 can be 
used to average over initial spin states and sum over final spin states. For 
example, the sums over spin states that result from R by R matrices and 
R by S plus R by S matrices are 



RR 



[ Sp[fa + m)y^ 2 + m)yJSp [fa + my^fa + m)y y 

[(fa -fa) 2 ] 2 



Sp [fa + m)y y fa + m)y„ fa + m)y v fa + m)y 



fa - fa) 2 fa - far 

By judicious use of the spur relations given in Lecture 23 the following dif- 
ferential cross section is obtained (alternatively, Table 13-1 could be used 
to calculate M directly): 



2e 4 p dft 



4x 2 + 8x cos + 2(1 - cos 2 9) + 4 cos 9 



(1 - cos 9) 

*fl) -4cosl 
(1 + COS0) 2 (l-cos0)(l + cos#)_ 



4x 2 - 8x cos 0+2(1- cos 2 9) -4 cos 9 4(1 + x)(x - 3) 



126 QUANTUM ELECTRODYNAMICS 

E,p 




FIG. 25-3 

where x = E 2 /p 2 - This is called Moller scattering (see Fig. 25-3). 

Problems: (1) Calculate positron-electron scattering by the pre- 
ceding method. 

(2) Find the cross section for a \x meson to produce a knock-on 
electron. Assume that the fi meson satisfies the Dirac equation with 
S = 1/2 and no anomalous moment. Remember that the particles are 
distinguishable and hence there is no interchange of particles. 

(3) Calculate the expected electron-proton scattering cross section 
assuming the proton has no structure but does have an anomalous 
moment. The Dirac equation for a proton is (see page 54) 

(if+ M-eJ)L -{n/4M)y u y v F^)^= Of 

Thus the perturbing potential can be taken as (see page 54) 

eJjL + (e M /4M)y M y y (V„ A v - V V A^) 

and the coupling with a photon is 

e^ + (e M /4M)(^ - e^) or ey^ + (e M /4M)(^y M -y^i) 

The Sum over Four Polarizations. In classical electrodynamics, longitu- 
dinal waves can always be eliminated in favor of transverse waves and an 
instantaneous Coulomb interaction. This is the approach used by Fermi (see 
Lecture 1), and it will now be demonstrated that the sum over four polariza- 
tions is also equivalent to transverse waves but plus an instantaneous Cou- 
lomb interaction. If instead of choosing space directions x, y, z, one direc- 
tion parallel to Q (photon momentum) and two directions transverse to Q 
are taken, the matrix element can be written 



t For the proton y, = 1.7896. 



INTERACTION OF SEVERAL ELECTRONS 127 



M/-i47re 2 = (u 4 y t u 2 )(l/q 2 )(u 3 y 1: u 1 ) - ( u 4 y Q U2)(l/q 2 )(u 3 'y Q u 1 ) 
" £ (u 4 y tr U2)(l/q 2 )(u 3 y tr u 1 ) 

2 tr. direc. 

where yg is the y matrix for the Q directions and y tr represents the y ma- 
trix in either of the transverse directions. The matrix element of ^ = q 4 y t 
- Qyp is zero in general (from the argument for gauge invariance). f Thus 
yg can be replaced by (q 4 /Q)y t with the result 

^2 = (u A y t u 2 )^ (l-^hu 3 y t Ui) - £ KTtrUa) ~2 (u a r tt Ui) 

= -(U 4 y t u 2 )^2 (Wt u l) ~Tj (^47tr u 2)l2 (u 3 y t rUi) 
^ 1,2 * 

Now 1/Q 2 represents a Coulomb field in momentum space and y t is the 
fourth component of the current density or charge, so that the first term 
represents a Coulomb interaction while the second term contains the inter- 
action through transverse waves. 



tin our special case, it is easy to see directly, for example, 

(U 4 ^U 2 ) = (U 4 $ 2 -^4)U 2 ) = (^4^2 U 2) " (U 4 ^4 U 2) 

= m(u 4 u 2 ) - m(u 4 u 2 ) = 



Discussion 

and Interpretation 

of Various 
"Correction" Terms 

Twenty -sixth Lecture 

In many processes the behavior of electrons in the quantum -electrody- 
namic theory turns out to be the same as predicted by simpler theories save 
for small "correction" terms. It is the purpose of the present lecture to 
point out and discuss a few such cases. 

ELECTRON-ELECTRON INTERACTION 

The simplest diagrams for the interaction are shown in Fig. 26-1. The 
amplitude for the process has been found to be proportional, in momentum 




128 



"CORRECTION" TERMS 129 

representation, to 

(u 3 r fi u 1 )(u 4 r 1L[ u 2 )/q 2 

where q = (Q, q 4 ) and Q is the momentum exchanged by the two electrons. 
Also, since sk = F^i ~ $i it follows that 

(l^Ui) = (U 8 (A - F^Ut) = 

From this identity it was deduced in the last lecture that the amplitude for 
the process as just given is equivalent to 

[-(u 3 y t ui)(u 4 y t u 2 )/Q 2 ]-^ (u 3 ytrUi)(u 4 y tr U2)/q 2 

1,2 

By taking the Fourier transform of the first term, it can be seen that it is 
the momentum representation of a pure, instantaneous Coulomb potential. 
The second term then constitutes a correction to the simple Coulomb inter- 
action. In it y tr denotes the y's for two directions transverse to the direc- 
tion of Q. 

For slow electrons, the correction to the Coulomb potential may be sim- 
plified and interpreted in a simple manner. Note that in this case 

Q =Pl -Ps 

and 

q 4 = Ei - E 2 « [m+ (Pi 2 /2m)] - [m + (p 3 2 /2m)] = ( Pl 2 -p 3 2 )/2m 

= [(Pi + p 3 )/2m](pi - p 3 )~ v(pi - p 3 ) 

so that q 4 2 ~ v 2 Q 2 and q 2 in the denominator can be replaced by -Q 2 with 
small error. (In the C.G. system, q 4 = exactly.) The correction term be- 
comes 

+ Tj (^3rtrU 1 )(u 4 y tr u 2 )/Q 2 

1,2 

but 

(WtrUi) = u 3 *a tr u 1 

It is recalled that u = I a J, where u a is the large part and u b the small 
part and that in the nonrelativistic approximation 

u b « (l/2m)(a • n)u a 



130 



QUANTUM ELECTRODYNAMICS 



Also, since 

-C':) 

it follows that (taken between positive energy states) 

= l/2m[u 3 * (7(a-n 1 )+(<7-n 3 )(7u la ] tr 

In free space II = p, so the x component, for example, of the foregoing ma- 
trix is 

M<*xPlx + °y Ply + u z Plz) + (°"xP3x + °y P3y + °z P3z )°x 
" (Pi + P3)x + i[cr z (Pl-Ps)y " Oy(Pi-Ps) x ] 

where the commutation relations for the cr's have been used. From this it 
is easily seen that the amplitude for the correction to the Coulomb potential 
may be written altogether in the form 



S<?H 



X J u 



l 4a 



P1+P3 _ t O-X (Pi -Ps ) 
2m 2m 



P4 + P2 _ . O-X (P2 ~ P4) 

2m 2m 



"la} 



U 2 ; 



tr 



The first terms in each of the brackets represent currents due to motion of 
the electron transverse to Q and the second terms represent the transverse 
components of the magnetic dipole of each. So altogether it appears that the 
correction arises from current-current, current-dipole, and dipole -dipole 
interactions between the electrons. These interactions are expected even on 
the basis of classical theory and were described by Breit before quantum 
electrodynamics, hence are referred to as the Breit interaction. 

Consider the dipole-dipole term arising in the correction factor. Since 
Q = Pi " P3 = P2 - P4 it; is 

E (<7l*Q)tr (CT 2 XQ)tr/Q 2 

1,2 

But since (jXQ is zero when a and Q have the same direction, the sum 
could as well be over all three directions and then it is equivalent to a dot 
product. That is, this term of the correction is 



fa XQ)- (cr 2 xQ)/Q^ 



CORRECTION" TERMS 131 



By taking the Fourier transform f this will be seen to be the momentum 
representation of the interaction between two dipoles as was stated. 

Note that the approximation q 4 ~ (v/c)Q used above applies only between 
positive energy states. For, if one of the states represents a positron, then 

q 4 = Ej - E 2 * 

= 2m 

However, 2m is very large, so the correction is still small. It is necessary 
to redo the analysis nevertheless. 

ELECTRON-POSITRON INTERACTION 

It would appear that, since the electron and positron are distinguishable, 
the Pauli principle would not require the interchange diagram, leaving as the 
only one Fig. 26-2. 




FIG. 26-2 



But it is still possible by the same phenomenological reasoning to con- 
ceive of the diagram in Fig. 26-3, which would represent virtual annihilation 
of the electron and positron with the photon later creating a new pair. It 
turns out that it is necessary to regard an electron-positron pair as exist- 
ing part of the time in the form of a virtual photon in order to obtain agree- 
ment with experiment. 



t Notice that (a^ x Q) • (<y 2 X Q) exp (-iQ • x), which will appear in trans- 
form integral, is the same as -(^i x V ) • (<r 2 * V) exp (-iQ ■ x), where V is 
the grad operator. This device enables an integration by parts, which greatly 
simplifies the process and the result. Thus, since the transform of 1/Q 2 is 
1/r, the coupling is -(oi x V) • (<t 2 x V)(l/r), which is the classical energy 
for interacting magnetic dipoles. 



132 QUANTUM ELECTRODYNAMICS 




FIG. 26-3 

From the point of view that positrons are electrons moving backward in 
time, Fig. 26-3 differs from Fig. 26-2 only in the interchange of the "final" 
states ^ 3 , ^ 4 . The Pauli principle extended to this case continues to oper- 
ate; the amplitudes of the two diagrams must be subtracted, since they dif- 
fer only in which outgoing (in the sense of the arrows) particle is which. 

POSITRONIUM 

An electron and positron can exist for a short time in a hydrogenlike 
bound state known as the atom positronium. The ground state of positronium 
is an S state and may be singlet or triplet, depending on the spin arrange- 
ment. As has been indicated in assigned problems, the *S state can anni- 
hilate only in two photons, whereas the 3 S state decays only by three -photon 
annihilation. The mean life for two-photon annihilation is 1/8 x 10 9 sec and 
for three photons it is 1/7 x 10 6 sec. 

Problem: Check the mean life 1/8 x 10 9 sec for two-photon anni- 
hilation using the cross section already computed and using hydrogen 
wave functions with the reduced mass for positronium. 

Figure 26-2 contributes the Coulomb potential holding the positronium 
together. The correction term (Breit's interaction) arising from this same 
diagram contributes a dipole-dipole or spin-spin interaction that is different 
in the 3 S and *S states (the current-current and spin-current interactions 
are the same for both states). Thus this amounts to a fine-structure sepa- 
ration of the 3 S and S states which can be shown to be 4.8 x 10 ev. 

In view of the fact that a photon has spin 1, and the *S state of positro- 
nium spin 0, conservation of angular momentum prohibits the process in Fig. 
26-3 from occurring in the *S state. It does occur in the 3 S state, however. 
The term arising from this diagram is small and, therefore, constitutes an- 
other fine-structure splitting of the 3 S and S levels. It can be shown to 



CORRECTION" TERMS 



133 



amount to 3.7 x 10 ev in the same direction as the spin-spin splitting. It 
is referred to as splitting due to the 'mew annihilation force." 

In order to calculate the term arising from Fig. 26-3, one needs to com- 
pute 

-(u 4 y JLl u 1 )(u3r |i U2)/q 2 



In this case q w 4m (Q = in the C.G. system), and all matrix elements 
are 1 or (regarding particles as essentially at rest in the positronium), 
so the result is just a number. This means that taking the Fourier transform 
one gets a 6 function of the relative coordinate of the electron and positron 
for the interaction in real space. For this reason it is sometimes referred 
to as the "short-range" interaction of the electron and positron. 

The combined fine -structure splitting due to the effects already outlined 
turns out to be represented by 



(1/2) a 2 Rydberg (7/3) 

where a is the fine -structure constant. This amounts to 2.044 x 10 5 Mc, 
using frequency as a measure of energy. 

There is still another correction, however, not yet mentioned, arising 
from diagrams, such as Fig. 26-4, where the electron or positron may emit 




FIG. 26-4 

and reabsorb its own photon. Taking this into account, the fine-structure 
splitting in positronium is given by f 

(1/2) a 2 Rydberg {(7/3) - [(32/9)+ 2 In 2] (a /it)} 



tPhys. Rev., 87, 848 (1952). 



134 



QUANTUM ELECTRODYNAMICS 



having a value of 2.0337 x 10 5 Mc. The experimental value for the positro- 
nium fine structure is 2.035 ± 0.003 Mc, so it is seen that this last correction, 
though of order a smaller than the main terms, is necessary to obtain 
agreement with experiment. It is referred to both in positronium and in hy- 
drogen as the Lamb-shift correction because of its experimental observa- 
tion by Lamb as the source of the small splitting between the 2 Sj/ 2 and 2 Pi/2 
levels in hydrogen. In general, it comes under the heading of self-action of 
the electron, to be treated in more detail later. 



TWO-PHOTON EXCHANGE BETWEEN ELECTRONS AND/ OR 
POSITRONS 

It is easy to imagine that processes, indicated by the diagrams in Fig. 
26-5, may occur where two photons instead of one are exchanged. Although 





FIG. 26-5 



it has not been necessary to consider such high-order processes to secure 
agreement with experiment, it may become necessary as experimental re- 
sults improve. The amplitudes for the processes may be written down easily 
but their calculation is difficult. The amplitude for case II in space -time 
representation is, for example, 

-e 4 f JJS [K + (3J)y v K + (7,5)y M K + (5,l)l [K + (4,8)r fi K + (8,6)y, 

x K + (6,2)] S + (s 2 7j6 )<5 + (s 2 5, 8 ) dT 5 dT 6 dT 7 dT 8 

or in momentum representation it is 

-(4^)V/(u 3 y, ^_ K 1 i _ m r M u 1 )(u 4 y ^ 2 _ K 1 2 _ m y t/ u 2 ) 

d*fe 



ki 2 k 2 2 (27r) 4 



CORRECTION" TERMS 



135 




FIG. 26-6 



where 



ii ~ ti + Ki = rf 4 



or 



% = A + Ki ~ & 



(see Fig. 26-6). Thus it is possible to determine #i and #2 in terms of each 
other but not independently; that is, the momentum may be shared in any 
ratio between the two photons. It is for this reason that the integral over #i 
arises in the expression for the amplitude. 



Twenty -seventh Lecture 

SELF-ENERGY OF THE ELECTRON t 

In Lecture 26 the following idea was introduced: An electron may emit 
and then absorb the same photon, as in Fig. 27-1. Then the propagation ker- 
nel for a free electron moving from point 1 to point 2 should include terms 
representing this possibility. Including only a first-order term (only one 
photon is emitted and absorbed), the resulting kernel is 

K(2,l) = K + (2,l) - ie 2 /J K + (2,4)y„K + (4,3)y„K + (3,l)6 + (s 4 , 3 2 ) 



x dT 4 dT 3 



(27-1) 



The correction term in this equation is written down by an inspection of 
the diagram, following the usual procedure for scattering processes. In the 
present case, the initial and final momenta are identical. Therefore the 



|R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. 



136 



QUANTUM ELECTRODYNAMICS 



nondiagonal elements in the perturbation matrix will all be zero. A diagonal 
element is one in which the resulting wave functions of a particle remain 
in the same eigenstate. For time -independent perturbations, it was shown in 
the development of perturbation theory that the only effect on such wave func- 
tions is a change in phase, proportional to the time interval T over which 
the perturbation is applied. The resulting wave function is 



exp (-iE n T) exp[-i(AE)T] 



(27-2) 



Since the perturbation effect (AE)T is small, the second exponential can 
be expanded as 1 - i(AE)T + ••• and higher-order terms neglected. It is the 
second term of this expansion which is represented by the integral on the 
right side of Eq. (27-1). The representation is not yet an equality, since 
certain normalizing factors are different in the two expressions. 




K + (3,l) 



FIG. 27-1 



To obtain the correct equation proceed as follows: First, it is clear that 
the probability of the occurrence depends only on the interval in space and 
time between points 3 and 4, and not at all on the absolute values of the 
space and time variables. So suppose a change of variable is made so that 
dT4 represents the element of interval (in space and time) between 3 and 4. 
Then write the integral in Eq. (27-1) 



//f(4)y M K + (4,3)y M 6 + (s 4)3 2 )f(3)dT 4 d 3 x 3 dt 3 



(27-3) 



where it is clear that the operators K + and 6+ depend only on the interval 
3-4. 

Second, expression (27-2) contains the time -dependent part of the wave 
function, exp (-iE n t), because it was assumed that the wave functions used 
did not contain time factors. In Eq. (27-3), f(3), f(4) do already include the 
time -dependent part, so it should be omitted in Eq. (27-2). 



"CORRECTION" TERMS 137 

Third, the normalization of wave functions is different for the two ap- 
proaches. For the development that led to Eq. (27-2), the normalization 

/*** dv = 1 

was used. For the present development the normalization is 

j u* u dv = (2E/cm 3 ) • V (27-4) 

Thus, to establish an equality, expression (27-3) must be divided by the nor- 
malizing integral of Eq. (27-4). 
The resulting expression is 



-ie 2 //f (4)y u K + (4,3)y u 6 + (s 4 , 3 2 )f(3) dr 4 d 3 x 3 dt 3 
2E-V 



The integral over d 3 x 3 gives a V which cancels with the denominator, and 
the integral over dt 3 gives a T which cancels with the left-hand side, so 
finally 

2E AE =+e 2 / uy M K + (4,3)y M 6 + (s 4)3 2 )u dr 4 (27-5) 

Note that the integral is relativistically invariant. Further, since p is 
the same before and after the perturbation and E 2 - m 2 + p 2 , the change in E 
can be taken as a change in the mass of the electron, from 

2E AE = 2m Am 

Using this expression, and transforming to momentum space, 

47re 2 r~( l \ d 4 k 1 /or7 ox 

Am = - — r J u y u -j — -, y u u -—7 72 (27-6) 

2mi J \ V p-K-m Vj (2iry k^ 

The integrand may be rewritten from 

1 y u (^-^ + m)y fJ _2_ m + 2 }L 



'Vtf-K-m'l 1 ^-2p-k + k'-m' i k'-2p-k 

using ^u = mu and the relations of Lecture 10. Then Eq. (27-6) becomes 

4?re 2 C 2m + 2]£ d 4 k 1 /97 RM 

Am = I —9 — t —9 (27-6') 

i J k 2 -2p-k (2tt) 4 k 2 V 

This integral is divergent, and this fact presented a major obstacle to 
quantum electrodynamics for 20 years. Its solution requires a change in the 
fundamental laws. Thus suppose that the propagation kernel for a photon is 



138 



QUANTUM ELECTRODYNAMICS 



(l/k 2 )c(k 2 ) instead of just (1/k 2 ), where c(k 2 ) is so chosen that c(0) = 1 and 
In space representation the modification takes the form 



c(k 2 )-~0 as k 



2_~« 



<5 + (si2 2 )-^f + (s 1)2 2 )= J(l/k 2 )c(k 2 )exp(-ik-x)d 4 k/(27r) 4 



(27-7) 

The new function f + differs significantly from 6 + only for small inter- 
vals. This is clear from the fact that if the high-frequency components are 
removed from the Fourier expansion of a function, only the short-range de- 
tails are modified. In the present case the size of the interval over which 
the function is modified can be described roughly as follows: Consider a 



large number, X , and suppose that so long as k « A.' 



c(k 2 ) 



1. Then (from 



the exponential term) differences will occur when the interval s^» 1/X . Call 




FIG. 27-2 



this value a , and the general behavior of f + is shown by Fig. 27-2. Thus a' 
is sort of a ''mean width" of f , . If a 2 « 1, as assumed, then when 



t 2 -r 2 



a 2 /2r 



(27-8) 



which is the size of the interval. The significance of the form of f + (s ) can 
be understood from the following. The original function, <5 + (s 2 ) differs from 
zero only when s = t — r = 0. That is to say, an electromagnetic signal can 



reach a point at distance r only at a time t such that t 2 - r 2 = or t = r 
(i.e., the speed of light is 1). This is no longer true for f + (s 2 ). The depar- 
ture is obtained by a measure of t - r. But, by Eq. (27-8), for all values of 
r »a this measure is negligible. Thus, depending on A 2 , the laws will be 
found unaffected over any practical distance. 



"CORRECTION" TERMS 139 

Choosing A 2 » m 2 , a practical (and general) representation of c(k 2 ) is 

c(k 2 ) = Jg(X) dX(-X 2 )(k 2 - A 2 )" 1 
and the simple form is suggested, 

c(k 2 ) = -X 2 /(k 2 - A 2 ) 
From this, obtain the propagation kernel as 

l/k 2 (-A 2 )(k 2 -A 2 ) _1 = 1/k 2 - l/(k 2 -A 2 ) 

The second term is that for the propagation of a photon of mass X; how- 
ever, the minus sign in front of the term has not been explained so far from 
this point of view. 

A convenient representation for this kernel is the integral 
r x 2 

-J o dL/(k 2 - L) 2 (27-9) 

Introducing this kernel into Eq. (27-6' ) in place of 1/k 2 gives 

f 2m + 2k djk/ -X 2 \ 
J k 2 - 2p • k k 2 U 2 - X 2 ) (27 " 10) 

which can be written as the sum of two integrals, which differ only by having 
m or ^ in the numerator, that is, m or k (since # = k y ). 

METHOD OF INTEGRATION OF INTEGRALS APPEARING IN 
QUANTUM ELECTRODYNAMICS 

We shall need to do many integrals of a form similar to the preceding 
one. A method has been worked out to do these fairly efficiently. We now 
stop to describe this method of integration. 

Everything will be based on the following two integrals:! 

roo (l;k )d 4 k _ 

J /0 \ n 2 . — 3 = (327T 2 iL)- 1 (l;0) (27-11) 

J -°° (27r) 4 (k^ + ie - L) d v ' v ' v ; 

J [ax + b(l -x)]" 2 dx = 1/ab (27-12) 

In Eq. (27-11), to write a little more compactly, we use the notation (l;k ) 
to mean that either 1 or k Q is in the numerator, in which case, on the right- 
hand side the (1;0) is 1 or 0, respectively. To prove the first of these, note 



•j-R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. Note 
that in the article d 4 k is equivalent to 47r 2 [d 4 k/(27r) 4 ] in our notation. 



140 QUANTUM ELECTRODYNAMICS 



that, if k is in the numerator, the integrand is an odd function. Thus the 
integral is zero. With 1 in the numerator, contour integration is employed, 
Write the integral 

Then for e « L + k 2 , there are poles at cu= ±[(L + k 2 ) l/2 - ie], and contour 
integration of lo gives 



J°° [(J + ie - (L + k 2 ) J" 1 dw = 2?ri[ -2(L + k 2 ) _l/2 ] 



with the contour in the upper half -plane. Two differentiations with respect 
to L give 

f_ °°Ju> 2 + ie - (L + k 2 )] " 3 da; = (67r/16i)(L + k 2 )" 5/2 

Then the remaining integral is 

//~/(L + k 2 )" 5/2 d 3 k = 4?r /°(L+k 2 r 5/2 k 2 dk 

= 4?r [k 3 /3L(L+ k 2 ) 3/2 ] | °° = 4?r/3L 

which proves Eq. (27-11). If k-p is substituted for the variable of integra- 
tion in Eq. (27-11), the result is 

r oo (l;k )d 4 k 

L (27r) V_ 2p . k _ A) 3 = I 32 - 2 HP 2 * A)l "' (l ; Po) (27-13) 

By differentiating both sides of Eq. (27-13) with respect to A or with respect 
to Pj , there follows directly 

roo (l;k ;k kj)d 4 k = _ [l;p ;p pj - (l/2)6 oJ (p 2 + A)] 

^-« (2tt) 4 (k 2 - 2p • k - A) 4 96tt 2 i(p 2 + A) 2 



Further differentiations give directly successive integrals including more k 

factors in t 

nominator. 



factors in the numerator and higher powers of (k 2 - 2p • k - A) in the de 



Twenty -eighth Lecture 

SELF-ENERGY INTEGRAL WITH AN EXTERNAL POTENTIAL 

Last time it was found that the self-energy of the electron is equivalent 
to a change in mass 



CORRECTION" TERMS 141 



Am "2ml j k 2 - 2p • k U 2 - A 2 J k 2 (27T) 4 (28 ~ lj 

and that this could also be expressed in terms of integrals, 

/A 2 , (l;k ) d 4 k 

1 = "Jo dL J (k 2 -2p-k)(k 2 -L) 2 ^ (28 " 2) 

It was also found that 

r (ljk ) <^ 9 12 1 

f (k 2 -2p-k-A) 3 = (32?r i} (P + A) (28 " 3) 

Using the definite integral 

JL _ f 1 _ 2(i-x)dx 

ab 2 " j o [ax + b(l-x)] 3 (28 4) 

the denominator of the integrand of Eq. (28-2) may be expressed as 



= 1 



2(1 -x) dx 



(k 2 - 2p • k)(k 2 - L) 2 J o [k 2 - 2xp • k - L(l - x)] 3 

so that Eq. (28-2) becomes 

A 2 , ,1 d 4 k(l;k G )2(l - x) dx 

I=-J dL J i [k 2 - 2xp • k - L(l - x)] 3 (2^ (28_5) 

The integral over k can be done by using Eq. (28-3) with the substitutions 
xp for p and L(l — x) for A, giving 



I = -X dL I 



X 2 ,1 (l;pJ2(l -x)dx 



2 2 



TTT73272TT7. ^T P =m 



'o ^o [327rM][x^p^+ L(l -x)] 

The integral over L is elementary and gives 

1 = -2(327r 2 i) _1 J dx(l;xp ) ln[(l-x)A 2 +m 2 x 2 /m 2 x 2 ] 

When A 2 »m 2 , it is legitimate to neglect m 2 x 2 in the numerator [it is true 
that when x~ 1, (1 - x)X 2 is not much larger than m 2 x 2 , but the interval 
over which this is true is so small, for A 2 »m 2 , that the error is small], 
so that, when the x integration is performed,! 



r 1 r 1 

t 1 In [x" 2 (l -x)] dx= 1 J x ln[x" 2 (l - x)] dx = -1/4 



142 QUANTUM ELECTRODYNAMICS 

I w -(327r 2 i)- 1 {2[ln(\ 2 /m 2 )+ 2]; p [ln(\ 2 /m 2 ) - 1/2]} 

X »m 2 

The change in mass is [from Eq. (28-1)] 

Am= (47r 2 /2mi)(-327r 2 i)- 1 (u{2m[2 ln(\ 2 /m 2 ) + 2] 

+ 2^[ln(A 2 /m 2 ) - (l/2)]}u) 

Since ^u = mu and (uu) = 2m, this can be simplified to 

Am/m = (e 2 /27r) [3 In (\/m) + (3/4)] (28-6) 

Now (e 2 /27r) is about 10~ 3 , so that even if A. is many times m, the fraction 
change in mass will not be large. The interpretation of this result is as fol- 
lows. There is a shift in mass which depends on k and hence cannot be de- 
termined theoretically. One can imagine an experimental mass and a theo- 
retical mass which are related by 

m exp =m th +Am < 28 " 7 ) 

All our measurements are of m , that is, self-action is included, and m th 
the mass without self-action, cannot be determined. More accurately stated, 



'A theory using m^ and 
e 2 /fic self-action 



{a theory using m exp , plus 
e 2 /hc self-action minus 
Am as computed for a 
free particle 



When the electron is free, the e 2 /fic self-action term exactly cancels the 
Am term and a theory using m exp is exactly correct. When the electron is 
not free, e 2 /Rc self-action is not quite equal to the Am term and there is 
a small correction to a theory using m ex p • This effect leads to the Lamb 
shift in the hydrogen atom, and, in order to calculate such effects, we shall 
now consider the effect of self-action on the scattering of an electron by an 
external potential. 

SCATTERING IN AN EXTERNAL POTENTIAL 

The diagram for scattering in an external potential is shown in Fig. 28-1, 
and the relationships for this process, excluding the possibility of self- 
action, are as follows: 

Potential: a'(q) = y t (47rZe/Q 2 )6(q4) for Coulomb potential 

Matrix element: M = -ie( u 2 ^ u i) 

Conservation relation: ^ 2 = K^i + & 



CORRECTION" TERMS 



143 




First-order self-action will produce the diagrams shown in Fig. 28-2. The 
amplitude for process is obtained in the usual manner. For example, dia- 
gram I gives 



Rationalizing the denominators and inserting the convergence factor, this 
becomes 



_ 47re 2 r (^27 u ^2 -K+m] £\$ x -K+m]y u ui) 
1 i j """"' (k 2 - 2p 2 • k)(k 2 - 2 Pl • k) 



-y 



k 2 -X 2 



x(2,r^ 



(28-8) 



This expression also happens to diverge for small photon momenta (k) (a 
result which has been called the "infrared catastrophe," but which has a 






144 QUANTUM ELECTRODYNAMICS 



clear physical interpretation, discussed later). Temporarily the k 2 under 
d 4 k will be replaced by (k 2 - A. 2 m i n ), where A. 2 m i n « m 2 , to make the inte- 
gral convergent. This is equivalent to cutting off the integral somewhere 
near k = \ m i n , and the physical interpretation is left to Lectures 29 and 30. 
To facilitate the integration over k, the following identity is used: 

x 2 
- h . (k 2 - L)- 2 dL 



mm 



k 2 "A 2 min " k 2 -A 2 

X 2 - A 2 
A min A 


-X 2 


(k 2 " A 2 min )(k 2 - X 2 ) ~ 
1 


k 2 -X 2 



k — X min 



2 ss ™2 \s \ 2 



since X » m » X m - m . This substitution produces integrals of the form 

r^ 2 f (l;k ;k k r )(27r)-«d«k 

j ^ 2 mi n J (k 2 - 2 Pl • k)(k 2 - 2p 2 • k)(k 2 -L) 2 

To evaluate these integrals, we make use of the identity 
(ab)- 1 ^ f dy/[ay+b(l-y)] 2 



so that 






dz. 



(k 2 - 2pj • k)(k 2 - 2p 2 • k) J o (k 2 - 2^ y • k) 2 

where $ y = y^j + (1 - y)^ 2 • Performing integrations in the order, k, L, y, 
and using the appropriate integrals in Eq. (28-6) gives as the matrix to be 
taken between states u 2 and Uj 



M,-* 



r 2 fln-^--lV 1 --^-V^an 
L \ Amin A tan 20/ 



27rL"\'"X m in tan 20 tan 20 

r e 

x I a tan a da 
J 



27T 



to «* - «) T^e + ri . 



(28-9) 



where r = ln( X/m) +9/4-2 In (m/A min ) and 4m 2 sin 2 = q . 

It is shown in Lecture 30 that diagrams II and III (Fig. 28-2) produce a 



contribution M 2 + M 3 = -(e /2ir)r£ , which just cancels a similar term in M 3 . 
When q i 
mated by 



When q is small, 6 » (q 2 ) 1/2 /2m, and the sum M^ + M 2 + M 3 can be approxi 



CORRECTION" TERMS 



145 



M 



e 

47T 



2M<*-««» + iS?-> 



\ ^min 8/ 



(28-10) 



The (gfeL - 4<fy can be written out 

(M - M = y^Jy (qj,a y - a^q y ) 

But q^ is the gradient operator so this can be written, in coordinate repre- 
sentation, 

yflv ^n A v - v v A u ) = + y^v Y \iv 

[see Eq. (7-1)] . Reference to page 54 shows that the effect of a particle's 
having an anomalous magnetic moment is to subtract a potential [xy^Jy F„ y 
from the ordinary potential i. = y^ A^ appearing in the Dirac equation. Since 
this is precisely what the first term of Eq. (28-10) does, one can say that 
this part of the self-action correction looks like a correction to the elec- 
tron's magnetic moment, so that 

Melee = (e/2m)[l+ (e 2 /27r)] 

Note that this result [and (28-9) and (28-10)] does not depend on the cutoff 
X, and hence X can now be taken to be infinity.! 



Twenty -ninth Lecture 

It has been shown that when a particle is scattered by a potential, the pri- 
mary effect is that of $., and that for diagram I (Fig. 28-2) a correction term 
arises which is 




FIG. 28-2 



t R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume, 



146 



QUANTUM ELECTRODYNAMICS 



sH'"e-')( i -i^i)*"»» 



x*. 



x i ol tan a 



tan 26 



i + 



87rm 



(<$ - ii) 



28 



sin 20 



.2 
27T 






It remains to show that the combined effect of diagrams II and III (Fig. 28-2), 

I 

II 

FIG. 28-2 

when considered along with the effect of the mass correction, is another 
correction term, 








-{e 2 /2ii)v£ 

just canceling the last term in the preceding expression. It is recalled that 
the necessity for considering the effect of the mass correction together with 
the self-action represented in diagrams I, II, and III is that the theory being 
developed must contain the experimental mass rather than the "theoretical" 
mass. 

Suppose that in the Dirac equation 

(if - m th )* = e$* 

m t h , the theoretical mass, is replaced by m - Am, where m is the experi- 
mental mass: then 



(if - m)# = e(4(+Am)* 

The mass correction Am is just a number, so that in momentum represen- 
tation it is a 6 function of momentum. Hence from the form of the foregoing 
equation, it is seen to behave like a potential with zero momentum and in- 
volves no matrices. Diagrammatically its effect may be represented as in 
Fig. 29-1. The minus sign is used because the effect of the mass correction 
Am is to be subtracted from the results obtained from diagrams I, II, and 
III (Fig. 28-2) alone. For diagram II the amplitude would appear to be 



CORRECTION 



TERMS 



147 





FIG. 29-1 
~ 1 /47re 2 f 1 1 

and for diagram II' (Fig. 29-1), 



d 4 k -A 
(2tt) 4 k 2 - X 2 



■u 2 ;([l/(|i r m)](Am)u 



But the part of the amplitude for diagram II (Fig 28-2) contained in the pa- 
rentheses is just Amu 1} so that II and II' seem to cancel. A similar result 
applied for diagrams III and III'. This is an error, however, arising from 
the fact that both of these amplitudes are infinite, owing to the factor j^-m 
in the denominator. Hence their difference is indeterminate. But by sub- 
tracting them properly it will be found that their difference does not vanish. 

The method proposed to accomplish this subtraction will, in fact, give 
the combined effect of the self-action and mass correction of both diagrams 
II and III and II' and III' . It is based on the fact that an electron is never 
actually free. An electron's history will have always involved a series of 
scatterings, as will its future. These scatterings will be considered as oc- 
curring at long but finite time intervals. It will be sufficient to calculate the 
effect of self-action and the mass correction between any two of these scat- 
terings, since the result will evidently be the same between each pair of 
them. Then, the effect will be accounted for simply by regarding a correc- 
tion, equal to that calculated for one of the intervals between scatterings, 
as being associated with the potential at each scattering (number of inter- 
vals equals number of scatterings). Then, considering a single scattering 
event as here, this correction to the potential represents all the effects of 
diagrams H, III, II' , and III' . 

For an electron which is not quite free, p 2 * m 2 exactly, but instead 

p 2 = m 2 (l + e) 2 



where 



148 



QUANTUM ELECTRODYNAMICS 



me = K/T 

by the uncertainty principle, and T is the interval between scatterings. 
Since T is large, e is a small quantity. Let $ = (1 + e)$ Q , where $ is the 
momentum of a free electron. 

If i. and # are the momentum representatives of the scattering poten- 
tials at a and b (any two scatterings), then the matrix of the amplitude to 
go from the initial state at a to the final state at b without any perturba- 
tions is 



p i - m * p 2 - m 2 ^ 2m 2 e 



up to terms of order €. With the perturbations of self-action and mass cor- 
rection, this matrix is 



■, 2 fi/ I I 1 , d 4 k -A 2 





(a) Without perturbation 



-Am 



(b) With perturbation of self-action 
and mass correction 



FIG. 29-2 

It is the value of this matrix compared to that of the unperturbed matrix 
which gives the desired correction term (see Fig. 29-2). 

Problem: Show that for two noncommuting (or commuting) oper- 
ators A and B, the following expansion is true: 



1 1 1^1 l^lr.1 

= — - — B— + — B — B — + 

A + B A A A A A A 



Using the result of the preceding problem, one can write 



CORRECTION" TERMS 149 



1 



tf-tf-m rf + etf -\L-m tf -)£-m tf -)£-m 

1 



x e& 



K^ - \i - m 



so that the foregoing matrix becomes 

• a 2 f i; tf + m 1 ^ + m , d* 



d 4 k -A 



2m z e 'M ^ -k-m ^ 2m z £ " k z -A 2 min k 2 - A 2 
l47re J **2m 2 ^ ^ -K-m Po ^o-K-m^ 1?^?~ 



1^ + m , / -A 2 \ u j± m . yf+m 
2m 2 Mi?^^J-^ 2m 2 e Am f^7 



i 



The first and last terms are identical, up to terms of order e, hence may be 
canceled. The integral in the second term has already been done essentially 
in computing diagram I (Fig. 28-2), except here ^ replaces &, $ it and $ 2 > 
so that <& = $2 ~ $1 ~ in this case and gives the result 

2tt p 2m 2 e m 2m 2 * 

To this order in e the ^'s in the numerator may be replaced by $ 's. It is 
also noted that since ^ u = mu, 

# +m) A # + m) = 2m 2 $ + m) 
so that the foregoing result may be written 

This is just -(e 2 /27r)r times the matrix for no perturbation. Hence the cor- 
rection term due to diagrams II, III, II', and III' is obtained simply by re- 
placing the scattering potential i. by — (e 2 /27r)r^, as was stated earlier. 

It should be noted that the difficulty in obtaining the proper subtraction of 
the self-action and mass corrections just clarified does not represent a 
"divergence" problem of quantum electrodynamics. It is a typical problem 
which could as well arise in nonrelativistic quantum mechanics if, for ex- 
ample, one chose some nonzero value as a reference of potential, that is, 
regarded a free electron as moving in a uniform nonzero potential. It may 
be easily verified that this would give rise to an "energy correction" for 
the free electron analogous to the mass correction involved here. Then in 



150 QUANTUM ELECTRODYNAMICS 



computing the amplitude for a scattering process where one used a "theo- 
retical energy" and subtracted the effect of the "energy correction," the 
difference of infinite terms would appear if one used free-electron wave 
functions. In this simple case the infinite term would, indeed, cancel upon 
proper subtraction but in principle the problem is the same as the present 
one. 

Finally, the complete correction term arising from self-action and mass 
correction is 



ei 

2ir 



2fto r *S--l)(l--^) + 8tane + -± 
\ A m in /\ tan 20/ tan 



tan 26 



x 
'o 



J a tan a da 



e 2 .,, ,,. 2d 



i + ^zziM- M) 



87rm sin 26 

RESOLUTION OF THE FICTITIOUS "INFRARED CATASTROPHE" 

From the correction term just determined, it is seen that, to order e 2 , 
the cross section for scattering of an electron with the emission of no pho- 
tons is 




■"si- 1 t-A* f n 7 dent 

\on A m i n / 



where <7 is the cross section for the potential i. only. This cross section 
diverges logarithmically as A. m i n — * 0, and it is this divergence which was 
formerly referred to as the "infrared catastrophe." 

This result, however, arises from the physical fact that it is impossible 
to scatter an electron with the emission of no photons. When the electron is 
scattered, the electromagnetic field must change from that of a charge mov- 
ing with momentum pj to that for momentum p 2 . This change of the field is 
necessarily accompanied by radiation. 

In the theory of brehmsstrahlung, it was shown that the cross section for 
emission of one low-energy photon is 

2 



e_ d^jti f gPl e __ ^P2' e \ do; 
a ° 7T 47T \ Pf q P2*<1 / w 



Problem: Show that the integral over all directions and the sum 
over polarizations of the foregoing cross section is 

o- = cr Q (2e 2 A)[l - (20/tan 26)] duj/co 

where sin 2 6 = -($ 2 ~ F^i) 2 /4m 2 . Thus the probability of emitting any photon 
between k = and k = K^ is 



CORRECTION" TERMS 151 



2e 



7r V tan 20 A w ' a ° f \ tan 20/ n A min 



which diverges logarithmically. 

Therefore, the dilemma of the diverging scattering cross section actually 
arises from asking an improper question: What is the chance of scattering 
with the emission of no photons? Instead, one should ask: What is the chance 
of scattering with the emission of no photon of energy greater than K^ ? 
For there will always be some very soft photons emitted. 

Then, effectively, what is sought in answer to the last question is the 
chance of scattering and emitting no photon, the chance of emitting one pho- 
ton of energy below K^ , and the chance of two and more photons below K m 
(but these terms are of order e 4 and higher and hence are neglected). 

Each of these terms is infinite, actually, but is kept finite temporarily 
by the artifice of the X m i n . Their sum, however, does not diverge, as may 
be seen by gathering the previous results and by writing 



Chance of scattering and emitting no photon of energy > K 

2 



= *o i i ; - ! 



pendent of A. min 
of order e 4 ) 



2e 2 / 20 \ . K m 

1 - ~ ~ In ; + (terms 

7T \ tan 20; A min 



^o 



°b 



2 \ / «„ \-i /terms independent \ 

1 - S! 2 In -a- 1 - -2M1 + of X min and of 
{ , K m A tan2 9 jj l orde »» 4 j 



This does not depend on X m i n and hence resolves the "infrared catastrophe." 
It has been shown by Bloch and Nordsieck that the same idea applies to all 
orders, t 

It is interesting that the largest term in the quantum-electrodynamic 
corrections to the scattering cross section, namely, 

-(2e 2 A) [1 - (20/tan 20)] In (m/K m ) 

may be obtained from classical electrodynamics, since such long wave- 
lengths are involved. The other terms have small effects. To date, the scat- 
tering experiments have been accurate enough to verify the existence of the 
large term but not accurate enough to verify the exact contributions of the 
smaller terms. Hence they do not provide a nontrivial test of quantum elec- 
trodynamics. 

These same considerations apply in any process involving the deflection 



fF. Bloch and A. Nordsieck, Phys. Rev., 52, 54 (1937) 



152 QUANTUM ELECTRODYNAMICS 



of free electrons. The best way to handle the problem is to calculate every- 
thing in terms of the A. m j n and then to ask only questions which can have a 
sensible answer as verified by the eventual elimination of the A. m i n . 

Problem: Prepare diagrams and integrals needed for the radia- 
tive corrections (of order e 2 ) to the Klein-Nishina formula. Do as 
much as possible and compare results with those of L. Brown and 
R. P. Feynman.f 



Thirtieth Lecture 

ANOTHER APPROACH TO THE INFRARED DIFFICULTY 

Instead of introducing an artificial mass, assume no weak photons con- 
tribute. Thus we must subtract from the previous results the contributions 
of all photons with momentum magnitude less than some number k »\. 
The previous result is 

i{l + (e 2 /27r)[2 In (m/X min - 1)(1 -20/tan 29)] + 9 tan 9 

r e 
+ (4/tan 29) J q y tan y dyj} (30-1) 

The term to be subtracted is 

(e 2 /27r) J q °y u (p 2 - J£ + m)(k 2 -2p 2 ■ k^" 1 rftfi -tf+ m) 

x (k 2 - 2 Pl -k!)" 1 y^ d 4 k/(k 2 -A 2 min ) (30-2) 

We assume k « Pi or p 2 , and neglect both K and the first two k 2 in this 
integral. Then using ^y^ = 2p^ - y^i, the integral is approximately 

-|2 
P2 M _ JBlt 



<Li f 

27T 2 J 



p 2 • k Pi * k. 



d 4 k 



k 2 - X 2 • 



(30-3) 



Then 



x = e 2 /27r{[l - (20/tan 2 9)] [2 In (2koA min - 1)] + [40/tan 29] 

r 26 
x [(1/20) J o (y/tan y) dy - 1]} (30-4) 

This is the term to be subtracted from expression (30-1). 
Using sin 2 = q 2 /4m 2 , for small q, Eq. (30-4) becomes 

x = (e 2 /27r) (2q 2 /3m 2 )[ln (2k A min ) - (5/6)] 



tPhys. Rev., 85, 231 (1952). 



CORRECTION" TERMS 



153 



Subtracting this from Eq. (30-1), also with q small, gives 

i {1 + (e 2 /47r)(4q 2 /3m 2 )[ln M/A min ) - (3/8) - ln(2koAmin) 
+ (5/6)]} (30-5) 

The last term is [In (M/2ko) + (11/24)]. 

EFFECT ON AN ATOMIC ELECTRON 

Consider the hydrogen atom with a potential V = e 2 /r and a wave func- 
tion 4>q(B.) exp (-iE t) = o (x fI ). Take the wave function to be normalized in 
the conventional manner. The effect of the self-energy of the electron is to 
shift the energy level by an amount 

AE = e 2 /0o(x2 J t 2 )r /i K + v (2,l)r f ,6 + (s 1) 2 2 )0o(xi,t 1 )d 3 x 1 d 3 x 2 dt 2 



-Am/ 0(x,t)0(x,t) d l 



(30-6) 



The first integral is written down from Fig. 30-1. The second is the free- 
particle effect as noted in previous lectures. The kernel K + v is not well 




FIG. 30-1 



enough determined to make exact calculation of this integral possible. An 
approximate calculation can be made with the form 

K + V (2,l) = £ expI-iEoflfc-tjj] ? n (x 2 )0 n (x 1 ) t 2 >-tj 



- similar sum over negative energies for t 2 < t t 
The photon propagation kernel can be expanded as 



154 QUANTUM ELECTRODYNAMICS 

<5+(si )2 2 ) = 4tt Jexp[-ik(t 2 -t 1 ) + ik(x 2 -x 1 )] d 3 k/2k(27r)~ 3 

t 2 >ti 

= 4?r j exp [+ ik(t 2 - t t ) + ik(x 2 -x^l d 3 k/2k(27r)~ 3 

t 2 <tj 

Using these expressions, Eq. (30-6) becomes 

AE = S/[Q! M exp(-iK-R)] 0n (E n +K-E )- 1 [a fi exp(iK'R)] n0 

+ n 

x d 3 k/47rk _ £ J[a u exp(-iK-R)] 0n (|E n |+ u+ Eq)" 1 

-n 

x [a^ exp (iK • R)] n0 d 3 k/47rk - (Am term) (30-7) 

This form implies the use of <fi* instead of and a 4 = 1, a lj2j3 = a. 

Another approach to the motion of an electron in a hydrogen atom is the 
following. Consider the electron as a free particle intermittently scattered 
by the Coulomb potential. The scatterings cause a phase shift in the wave 
function of the order of (Rydberg/R ). Thus the period between scatterings 
is of the order T = K/Rydberg. Take the lower limit k of the momentum of 
the "self-action" photons as very large compared to the Rydberg. Then it 
is very probable that an emitted photon will be reabsorbed before two inter- 
actions between the electron and the potential have taken place; it is very 
improbable for two or more scatterings to take place between emission and 
absorption (see Fig. 30-2). Then the correction to the potential is that com- 
puted in Eq. (30-5) for small q (plus anomalous moment correction). This 
is 

(e 2 /47r)(4q 2 /3m 2 )(ln m/2k + 11/24) f 
in momentum space. To transform to ordinary space, use 

q 2 V = (q 4 2 - Q 2 ) t — (9 2 /9t 2 - V 2 ) V 

Thus the correction is 

-(e 2 /37rm 2 )(log M/2k + 11/24) V 2 V (30-7') 

This correction is of greatest importance for the s state, since with a Cou- 
lomb potential V 2 V = 47rZe 2 S(R), and only in the s states is 0(R) different 
from at R= 0. 

The choice of ko is determined by the inequalities m »k » Rydberg. A 
satisfactory value is k = 137 Ryd. With such a k , the effect of photons of 
k < k must be included. This will be done by separating the effect into the 
sum of three contributing effects. It will be seen that two of these effects 



CORRECTION" TERMS 



155 




k » Rydberg 
probable 




improbable 



FIG. 30-2 



are independent of the potential V and thus are canceled by similar terms 
in the Am correction for a free particle. Thus for only one situation must 
the effect be computed. In all cases, since k is small, the nonrelativistic 
approximation to expression (30-7) may be used. 

(1) The contribution of negative energy states: Neglecting k with respect 
to m gives 

(|E n | +k+ E ) « 2m 

The matrix element for a 4 is very small, and only the elements for a need 
be considered. Then the sum over negative states is 



S/W«0n)' (O!n0)/2m]k 2 dk/k 



If this sum is continued for +n, a negligible term of order v 2 /c 2 is added. 
Thus the sum is approximately 

"Si t(«0n) ' Ko)/2m] k 2 dk/k = (a ■ a) 00 k 2 dk/2mk 

all states 

= 3k 2 /4m 

This is independent of V, and thus is canceled by a similar quantity in the 
Am term. 

(2) Longitudinal positive energy states (a u —* Ot • k/k): As an exercise 
the reader may show 



a • k exp (ik • R) = H exp (ik ■ R) - exp (ik ■ R)H 



156 QUANTUM ELECTRODYNAMICS 

Then 

[(a • k/k) exp (ik • R)] n0 = (E n - E )/k[exp (ik • R)] n0 
and the contribution of these terms summed over positive energy states gives 
/ [1 - (E n - E ) 2 /k 2 ] exp (ik • R) 0n exp (-ik • R) n0 (E n + k - Eq)' 1 d 3 k/47rk 

= J(E n - E + k) exp (ik • R) 0n exp (-ik ■ R) n0 d 3 k/47rk 3 
= /[H exp (ik • R) - exp (ik • R)H] 0n [exp - (ik • R)] n0 d 3 k/47rk 3 
Writing H = p 2 /2m (V commutes with the exponent), this becomes 
f[(p + k) 2 /2m - p 2 /2m + k] d 3 k/k 3 

This term is independent of V, and thus is also canceled by the Am correc- 
tion. 

(3) Transverse positive energy states: Since k is large compared to the 
size of the atom, the dipole approximation can be used.f The general term 
in the sum of Eq. (30-7) becomes 

J(a tT ) 0n (a tT ) n0 (E n + k - Eq)" 1 d 3 k/k (30-8) 

Writing 

(E^k-Eo)- 1 ^ 1/k- (E n -E )/(E n +k-E )k 

the term in 1/k can be split off from the rest of the integral as a quantity 
independent of V and thus canceled by the Am correction. Further, by 
averaging over directions, 

Kr)0n (Oitr.)n0 = 2/3(a) 0n ' (O) n0 = (2/3m 2 )(p) 0n ' (p) n0 

in the nonrelativistic approximation. Thus the integral of Eq. (30-8) is 
(2/3m 2 )(p) 0n • (p) n0 (E n - E ) log (k + E n - E )/(E n - E ) 

Using the relation 

Pn o(E n -E ) = (pH-Hp) n0 = (VV) n0 



|Cf. H. Bethe, Phys. Rev., 72, 339 (1947), 



"CORRECTION" TERMS 157 

and the fact that k$ » E n - E , one part of the sum over transverse positive 
energy states is 

In k £Pon- (VV) n0 = 1/2 In k (V 2 V) o 

n 

This cancels with the In ko of Eq.(30-7'), leaving the final correction as 

(2e 2 /37rm 2 ) £ p n0 'Pon^ - E ){ log [M/2(E n - E )] + (11/24)} 

+ n 

+ anomalous moment correction 

This sum has been carried out numerically to be compared with the observed 
Lamb shift. 



Thirty -first Lecture 

CLOSED-LOOP PROCESSES, VACUUM POLARIZATION 

Another process which is still of first order in e 2 has not been consid - 
ered in the scattering by a potential. Instead of the potential scattering the 
particle directly, it can do so by first creating a pair which subsequently 
annihilates, creating a photon which does the scattering. Diagram I (Fig. 
31-1) applies to this process; diagram II applies to a similar process, with 
the order in time changed slightly. The amplitude for these processes is 

spin states ** \ *• n. i 

ofu (31_1) 

where u is the spinor part of the closed-loop wave function. The first pa- 
renthesis is the amplitude for the electron to be scattered by the photon; 
1/q 2 is the photon propagation factor; and the second parenthesis is the am- 
plitude for the closed-loop process which produces the photon. The expres- 
sion is integrated over p because the amplitude for a positron of any mo- 
menta is desired. In the sum over four spin states of u, two states take 
care of the processes of diagram I and two states take care of the proc- 
esses of diagram II. No projection operators are required, so the method of 
spurs may be used directly to give 



i47re 2 (uzT^Uj)-? /s P 



1 

7u 



q Lp-m n tf+d-m 



-4 j4 



(27T)- 4 d 4 p (31-2) 



a form which contains both I and II (so as usual it is not necessary to make 
separate diagrams for processes whose only difference is the order in time). 



158 



QUANTUM ELECTRODYNAMICS 



This integral also diverges, but a photon convergence factor, as used in the 
previous lectures, is of no value because now the integral is over p, the mo- 
mentum of the positron in the intermediate step. The method which has been 
used to circumvent the divergence difficulty is to subtract from this integral, 
a similar integral with m replaced by M. M is taken to be much larger 



closed 
loop 




f*2 = tfi + i 



closed 
loop 



i>2 = i>\~& 




than m, and this results in a type of cutoff in the integral over p. When 
this is done, the amplitude is found to be f 

(u 2 ynUi)a u (e 2 A)[-(l/3)ln(M/m) 2 - (1 - 0/tan 9) 



x (4m 2 + 2q 2 )/3q 2 + 1/9] 



(31-3) 



tSee R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. 



1 'CORRECTION" TERMS 159 

where q 2 = 4m 2 sin 2 6, which, for small q, becomes 

(Z 2 y ll u i )a ll (e 2 /7T)[-(l/3)ln(M/m) 2 + 2q 2 /l5] (31-4) 

Notice that (u 2 y JL( u 1 ) = (u 2 ^u 1 ), so that, considering only the divergent part 
of the correction, the effective potential is 

i {1 + (e 2 /7T)[- (1/3) In (M/m) 2 ]} (31-5) 

The 1 comes from the theory without radiative corrections, while the e 2 
term is the correction due to processes of the type just described. Thus 
the correction can be interpreted as a small reduction in the effect of all 
potentials, and one can introduce an experimental charge e exp and a theo- 
retical charge e^ related by 

eexp^th+Ae (31-6) 

where A(e 2 ) = -(e 2 /37r) In (M/m) 2 , in a manner analogous to the mass cor- 
rection described in Lecture 28. This is referred to as "charge renormal- 
ization." The other term, 

(2/15)(e 2 /7T)qV 

is more interesting, since it represents a perturbation 2e 2 /157r (V 2 V). This 
correction is responsible for 27 Mc in the Lamb shift and the {in [m/2(E n -E )] 
+ (11/24)} term in (30-7') is replaced by {in [m/2(E n - E )] + (11/24) -(1/5)}. 
The 1/5 term is due to the "polarization of the vacuum." 

SCATTERING OF LIGHT BY A POTENTIAL 

One possible process for the scattering of light, and an indistinguishable 
alternative, is indicated by the diagrams in Fig. 31-2. The second diagram 
differs from the first only in the direction of the arrows of the electron lines. 
Reversing such a direction is equivalent to changing an electron to a posi- 
tron. Thus the coupling with each potential would change sign. Since there 
are three such couplings, the amplitude for the second process is the nega- 
tive of that for the first. Since the amplitudes add, the net amplitude is zero. 
In general, any closed-loop process of this type involving an odd number of 
couplings to a potential (including photon), has zero net amplitude. 

Problem: Set up the integrals for each of the two diagrams in Fig. 
31-2 and show that they are equal and opposite in sign. 

However, the higher-order processes shown in Fig. 31-3 can take place. 
The amplitude for the process is 



160 




QUANTUM ELECTRODYNAMICS 

K 2 




FIG. 31-2 




-til f* ^ 'til 

alternatives 



V2 







FIG. 31-3 



< 'CORRECTION " TERMS 161 

-(47re 2 ) 2 /Sp WtdJ - m)" 1 fafa -fa - m)" 1 ^ -fa-fa- m)" 1 

x^ 4 (^ + *-m)- 1 ] (27r)" 4 d 4 k 

plus five similar terms resulting from permuting the order of photons. This 
integral appears to diverge logarithmically. But when all six alternatives 
are taken into account, the sum leaves no divergent term. More complicated 
closed-loop processes are convergent. 



Pauli Principle 

and the Dirac Equation 

In Lecture 24 the probability of a vacuum remaining a vacuum under the 
influence of a potential was calculated. The potential can create and anni- 
hilate pairs (a closed-loop process) between times tj and t 2 . The amplitude 
for the creation and annihilation of one pair is (to first nonvanishing order) 

L ~ //sp [K + (l,2)^(2)K + (2, l)rf(l)] dn dr 2 

The amplitude for the creation and annihilation for two pairs is a factor L 
for each, but, to avoid counting each twice when integrating over all dri and 
dT2, it is L 2 /2. For three pairs the amplitude is L 3 /3!. The total amplitude 
for a vacuum to remain a vacuum is, then, 

c v = 1 - L+ L 2 /2! - L 3 /3! + -" = e" L (31-7) 

where the 1 comes from the amplitude to remain a vacuum with nothing 
happening. The use of minus signs for the amplitude for an odd number of 
pairs can be given the following justification in terms of the Pauli principle. 
Suppose the diagram for t < t t is as shown in Fig. 31-4. The completion of 
this process can occur in two ways, however (see Fig. 31-5). The second 
way can be thought of as obtained by the interchange of the two electrons, 
hence the amplitude of the second must be subtracted from that of the first, 





FIG. 31-4 
162 



PAULI PRINCIPLE AND DIRAC EQUATIONS 



163 





FIG. 31-5 

according to the Pauli principle. But the second process is a one -loop proc- 
ess, whereas the first process is a two-loop process, so it can be concluded 
that amplitudes for an odd number of loops must be subtracted. The prob- 
ability for a vacuum to remain a vacuum is 



vac- vac 



|c v | 2 = exp (-2 real part of L) 



The real part of L (R.P. of L) may be shown to be positive, so it is clear 
that terms of the series must alternate in sign in order that this probability 
be not greater than unity. 

We have, therefore, two arguments as to why the expression must be 
e" L . One involves the sign of the real part, a property just of K + and the 
Dirac equation. The second involves the Pauli principle. We see, therefore, 
that it could not be consistent to interpret the Dirac equation as we do un- 
less the electrons obey Fermi-Dirac statistics. There is, therefore, some 
connection between the relativistic Dirac equation and the exclusion princi- 
ple. Pauli has given a more elaborate proof of the necessity for the exclu- 
sion principle but this argument makes it plausible. 

This question of the connection between the exclusion principle and the 
Dirac equation is so interesting that we shall try to give another argument 
that does not involve closed loops. We shall prove that it is inconsistent to 
assume that electrons are completely independent and wave functions for 
several electrons are simply products of individual wave functions (even 
though we neglect their interaction). For if we assume this, then 



Probability of vacuum 
remaining a vacuum 



Probability of vacuum 
to 1 pair 

Probability of vacuum 
to 2 pairs 



PV E iKlpairl 2 
all pairs 



P V E l K lpair| 2 iKlpairl 2 
all pairs 



164 QUANTUM ELECTRODYNAMICS 

Now, the sum of these probabilities is the probability of a vacuum becoming 
any thing and this must be unity. Thus 

1 = P v [1 + (prob. of 1 pair) + (prob. of 2 pairs) + • • •] (31-8) 

The probability that an electron goes from a to b and that nothing else hap- 
pens is P V |K + (b,a)| 2 . The probability that the electron goes from a to b 
and one pair is produced is P v |K + (b,a)| 2 |K(1 pair)| 2 , and the probability that 
the electron goes from a to b with two pairs produced is P v |K + (b,a)| 2 
x |K(2 pair)| 2 . Thus the probability for an electron to go from a to b with 
any number of pairs produced is 

P v |K + (b,a)| 2 [l + |K(lpair)| 2 + |K(2pairs)| 2 + . ■ . = |K + (b,a)| 2 

(31-9) 

[see Eq. (31-8)] . Now since the electron must go somewhere, 

J|K + (b,a)| 2 db=l 

However, it is a property of the Dirac kernel that 



/|K + (b, 



a)| 2 db> 1 (31-10) 



and an inconsistency results. The inconsistency can be eliminated by assum- 
ing that electrons obey Fermi-Dirac statistics and are not independent. Un- 
der these circumstances the original electron and the electron of the pair 
are not independent and 



Probability of electron from , 

4. u i i • ^ j r < K+(b,a)r K(l pair) ' 

a to b plus 1 pair produced ' 



(31-11) 



because we should not allow the case that the electron in the pair is in the 
same state as the electron at b. 

For the kernel of the Klein-Gordon equation, it turns out that the sign of 
the inequality in Eq. (31-10) is reversed. Therefore, for a spin-zero parti- 
cle neither Fermi-Dirac statistics nor independent particles are possible. 
If the wave functions are taken symmetric (charges reversed add ampli- 
tudes, Einstein-Bose statistics), the inequality Eq. (31-11) is also reversed. 
In symmetrical statistics the presence of a particle in a state (say 6) en- 
hances the chance that another is created in the same state. So the Klein- 
Gordon equation requires Bose statistics. 

It would be interesting to try to sharpen these arguments to show that the 
difference between J |K + (b,a)| 2 db and 1 is quantitatively exactly compen- 
sated for by the exclusion principle. Such a fundamental relation ought to 
have a clear and simple exposition. 



165 



AN OPERATOR CALCULUS 



123 



10. SUMMARY OF NUMERICAL FACTORS FOR 
TRANSITION PROBABILITIES 

The exact values of the numerical factors appearing 
in the rules of II for computing transition probabilities 
are not clearly stated there, so we give a brief summary 
here. 20 

The probability of transition per second from an 
initial state of energy £ to a final state of the same total 
energy (assumed to be in a continuum) is given by 

(h=c=l), 

Prob. trans/sec = 27riV- 1 |3H| 2 p(£), 

where p(E) is the density of final states per unit energy 
range at energy E and 1 9TI | 2 is the square of the matrix 
element taken between the initial and final state of the 
transition matrix 9TC appropriate to the problem. N is a 
normalizing constant. For bound states conventionally 
normalized it is 1. For free particle states it is a product 
of a factor Ni for each particle in the initial and for 
each in the final energy state. Ni depends on the 
normalization of the wave functions of the particles 
(photons are considered as particles) which is used in 
computing the matrix element of 2(11. The simplest rule 
(which does not destroy the apparent covariance of 
911), is 21 A r ,= 2ei, where e,- is the energy of the particle. 
This corresponds to choosing in momentum space, plane 
waves for photons of unit vector potential, e 2 = — 1. 
For electrons it corresponds to using {uu) = 2m (so that, 
for example, if an electron is deviated from initial ^ t to 
final p 2 , the sum over all initial and final spin states of 
|3TZ| 2 is Sp[(p t +m)m(pi+m)ml). Choice of norma- 
lization (uy t u)=?l results in A r ,= l for electrons. The 
matrix 311 is evaluated by making the diagrams and 
following the rules of II, but with the following defini- 
tion of numerical factors. (We give them here for the 
special case that the initial, final, and intermediate 

20 In I and II the unfortunate convention was made that d*k 
means dk i dkidk 2 dk i (2Tr)~ 2 for momentum space integrals. The 
confusing factor (2*)^ here serves no useful purpose, so the con- 
vention will be abandoned. In this section d*k has its usual meaning, 
dk<dkydk*lk 3 . 



states consist of free particles. The momentum space 
representation is then most convenient.) 

First, write down the matrix directly without 
numerical factors. Thus, electron propagation factor 
is (p—m)~ l , virtual photon factor is k~ 2 with couplings 
TV • "y> A real photon of polarization vector e M con- 
tributes factor e. A potential (times the electron charge, 
e) A^(x) contributes momentum q with amplitude a(q), 
where a»{q) = fA^X) exp(iq-xi)d 4 xi. (Note: On this 
point we deviate from the definition of a in I which is 
there (27r)~ 2 times as large.) A spur is taken on the 
matrices of a closed loop. Because of the Pauli principle 
the sign is altered on contributions corresponding to an 
exchange of electron identity, and for each closed loop. 
One multiplies by {2ir)^d i p={2ir)~ i dp l dp J (lp v dp z and 
integrates over all values of any undetermined mo- 
mentum variable p. (Note: On this point we again 
differ. 20 ) 

The correct numerical value of 91X is then obtained 
by multiplication by the following factors. (1) A factor 
(4ir)*e for each coupling of an electron to a photon. 
Thus, a virtual photon, having two such couplings, 
contributes 47re 2 . (In the units here, e 2 = 1/137 approxi- 
mately and (47r)*e is just the charge on an electron in 
heaviside units.) (2) A further factor —i for each virtual 
photon. 

For meson theories the changes discussed in II, 
Sec. 10 are made in writing 2flZ, then further factors are 
(1) (47r)*g for each meson-nucleon coupling and (2) a 
factor —i for each virtual spin one meson, but -\-i for 
each virtual spin zero meson. 

This suffices for transition probabilities, in which 
only the absolute square of 9TC is required. To get 311 
to be the actual phase shift per unit volume and time, 
additional factors of i for each virtual electron propa- 
gation, and — i for each potential or photon interaction, 
are necessary. Then, for energy perturbation problems 
the energy shift is the expected value of i3Tl for the 
unperturbed state in question divided by the normal- 
ization constant Ni belonging to each particle compris- 
ing the unperturbed state. 

The author has profited from discussions with 
M. Peshkin and L. Brown. 



21 In general, A, is the particle density. It is Ni=(uytu) for 
spin one-half fields and i[(<j>*d<j>/dt)-<t>d<j>*/dtl for scalar fields. 
The latter is 2a if the field amplitude <t> is taken as unity. 



167 



PHYSICAL REVIEW 



VOLUME 76 



MBER 6 



SEPTEMBER 15 



The Theory of Positrons 

R. P. Feynman 

Department of Physics, Cornell University, Ithaca, New York 

(Received April 8, 1949) 



The problem of the behavior of positrons and electrons in given 
external potentials, neglecting their mutual interaction, is analyzed 
by replacing the theory of holes by a reinterpretation of the. solu- 
tions of the Dirac equation. It is possible to write down a complete 
solution of the problem in terms of boundary conditions on the 
wave function, and this solution contains automatically all the 
possibilities of virtual (and real) pair formation and annihilation 
together with the ordinary scattering processes, including the 
correct relative signs of the various terms. 

In this solution, the "negative energy states" appear in a form 
which may be pictured (as by Stuckelberg) in space-time as waves 
traveling away from the external potential backwards in time. 
Experimentally, such a wave corresponds to a positron approach- 
ing the potential and annihilating the electron. A particle moving 
forward in time (electron) in a potential may be scattered forward 
in time (ordinary scattering) or backward (pair annihilation). 
When moving backward (positron) it may be scattered backward 



in time (positron scattering) or forward (pair production). For 
such a particle the amplitude for transition from an initial to a 
final state is analyzed to any order in the potential by considering 
it to undergo a sequence of such scatterings. 

The amplitude for a process involving many such particles is 
the product of the transition amplitudes for each particle. The 
exclusion principle requires that antisymmetric combinations of 
amplitudes be chosen for those complete processes which differ 
only by exchange of particles. It seems that a consistent interpre- 
tation is only possible if the exclusion principle is adopted. The 
exclusion principle need not be taken into account in intermediate 
states. Vacuum problems do not arise for charges which do not 
interact with one another, but these are analyzed nevertheless in 
anticipation of application to quantum electrodynamics. 

The results are also expressed in momentum-energy variables. 
Equivalence to the second quantization theory of holes is proved 
in an appendix. 



1. INTRODUCTION 

THIS is the first of a set of papers dealing with the 
solution of problems in quantum electrodynamics. 
The main principle is to deal directly with the solutions 
to the Hamiltonian differential equations rather than 
with these equations themselves. Here we treat simply 
the motion of electrons and positrons in given external 
potentials. In a second paper we consider the interactions 
of these particles, that is, quantum electrodynamics. 

The problem of charges in a fixed potential is usually 
treated by the method of second quantization of the 
electron field, using the ideas of the theory of holes. 
Instead we show that by a suitable choice and inter- 
pretation of the solutions of Dirac's equation the prob- 
lem may be equally well treated in a manner which is 
fundamentally no more complicated than Schrodinger's 
method of dealing with one or more particles. The vari- 
ous creation and annihilation operators in the conven- 
tional electron field view are required because the 
number of particles is not conserved, i.e., pairs may be 
created or destroyed. On the other hand charge is 
conserved which suggests that if we follow the charge, 
not the particle, the results can be simplified. 

In the approximation of classical relativistic theory 
the creation of an electron pair (electron A , positron B) 
might be represented by the start of two world lines 
from the point of creation, 1. The world lines of the 
positron will then continue until it annihilates another 
electron, C, at a world point 2. Between the times h 
and t 2 there are then three world lines, before and after 
only one. However, the world lines of C, B, and A 
together form one continuous line albeit the "positron 
part" B of this continuous line is directed backwards 
in time. Following the charge rather than the particles 
corresponds to considering this continuous world line 



as a whole rather than breaking it up into its pieces. 
It is as though a bombardier flying low over a road 
suddenly sees three roads and it is only when two of 
them come together and disappear again that he realizes 
that he has simply passed over a long switchback in a 
single road. 

This over-all space-time point of view leads to con- 
siderable simplification in many problems. One can take 
into account at the same time processes which ordi- 
narily would have to be considered separately. For 
example, when considering the scattering of an electron 
by a potential one automatically takes into account the 
effects of virtual pair productions. The same equation, 
Dirac's, which describes the deflection of the world line 
of an electron in a field, can also describe the deflection 
(and in just as simple a manner") when it is large enough 
to reverse the time-sense of the world line, and thereby 
correspond to pair annihilation. Quantum mechanically 
the direction of the world lines is replaced by the 
direction of propagation of waves. 

This view is quite different from that of the Hamil- 
tonian method which considers the future as developing 
continuously from out of the past. Here we imagine the 
entire space-time history laid out, and that we just 
become aware of increasing portions of it successively. 
In a scattering problem this over-all view of the com- 
plete scattering process is similar to the 5-matrix view- 
point of Heisenberg. The temporal order of events dur- 
ing the scattering, which is analyzed in such detail by 
the Hamiltonian differential equation, is irrelevant. The 
relation of these viewpoints will be discussed much more 
fully in the introduction to the second paper, in which 
the more complicated interactions are analyzed. 

The development stemmed from the idea that in non- 
relativistic quantum mechanics the amplitude for a 
given process can be considered as the sum of an ampli- 



749 



168 



750 



R. P. FE YN MAN 



tude for each space-time path available. 1 In view of the 
fact that in classical physics positrons could be viewed 
as electrons proceeding along world lines toward the 
past (reference 7) the attempt was made to remove, in 
the relativistic case, the restriction that the paths must 
proceed always in one direction in time. It was dis- 
covered that the results could be even more easily 
understood from a more familiar physical viewpoint, 
that of scattered waves. This viewpoint is the one used 
in this paper. After the equations were worked out 
physically the proof of the equivalence to the second 
quantization theory was found. 2 

First we discuss the relation of the Hamiltonian 
differential equation to its solution, using for an example 
the Schrodinger equation. Next we deal in an analogous 
way with the Dirac equation and show how the solu- 
tions may be interpreted to apply to positrons. The 
interpretation seems not to be consistent unless the 
electrons obey the exclusion principle. (Charges obeying 
the Klein-Gordon equations can be described in an 
analogous manner, but here consistency apparently 
requires Bose statistics.) 3 A representation in momen- 
tum and energy variables which is useful for the calcu- 
lation of matrix elements is described. A proof of the 
equivalence of the method to the theory of holes in 
second quantization is given in the Appendix. 

2. GREEN'S FUNCTION TREATMENT OF 
SCHRO DINGER'S EQUATION 

We begin by a brief discussion of the relation of the 
non-relativistic wave equation to its solution. The ideas 
will then be extended to relativistic particles, satisfying 
Dirac's equation, and finally in the succeeding paper to 
interacting relativistic particles, that is, quantum 
electrodynamics. 

The Schrodinger equation 



id$/dt = H$, 



(1) 



describes the change in the wave function \f/ in an 
infinitesimal time At as due to the operation of an 
operator exp( — iHAt). One can ask also, if ^(x x , h) is 
the wave function at Xi at time t\, what is the wave 
function at time t 2 >ti? It can always be written as 



/ 



iA(x 2 , h) = K(x it h; xi, *i)iKxi, h)dhL X 



(2) 



where if is a Green's function for the linear Eq. (1). 
(We have limited ourselves to a single particle of co- 
ordinate x, but the equations are obviously of greater 
generality.) If H is a constant operator having eigen- 
values E„, eigenfunctions 4> n so that ^(x, t\) can be ex- 
panded as £n C„0 n (x), then ^(x, / 2 ) = exp(— iE n {t 2 — h)) 
XC„</>„(x). Since C n = f<j> n * {^1)^1, h)d z x h one finds 

1 R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). 

'The equivalence of the entire procedure (including photon 
interactions) with the work of Schwinger and Tomonaga has been 
demonstrated by F. J. Dyson, Phys. Rev. 75, 486 (1949). 

3 These are special examples of the general relation of spin and 
statistics deduced by W. Pauli, Phys. Rev. 58, 716 (1940). 



(where we write 1 for Xi, h and 2 for x 2 , t 2 ) in this case 
K(2, l) = Z 0»(x 2 )«»*(x 1 ) exp(-iE n (t 2 -h)), (3) 

for ta>k- We shall find it convenient for h<h to define 
K{2, 1) = (Eq. (2) is then not valid for t 2 <h). It is 
then readily shown that in general K can be defined by 
that solution of 

(id/dt 2 -H 2 )K(2,l) = id(2,l), (4) 

which is zero for t 2 <t h where 5(2, l) = d(l 2 — h)8(x 2 — x{) 
X5(y 2 —yi)5(z 2 —Zi) and the subscript 2 on H 2 means 
that the operator acts on the variables of 2 of if (2, 1). 
When H is not constant, (2) and (4) are valid but K is 
less easy to evaluate than (3). 4 

We can call K{2, 1) the total amplitude for arrival 
at x 2 , t 2 starting from Xi, t\. (It results from adding an 
amplitude, expz'S, for each space time path between these 
points, where 5 is the action along the path. 1 ) The 
•transition amplitude for finding a particle in state 
x(x 2 , ti) at time t 2 , if at h it was in ^(xi, ti), is 



/ 



X*(2)K(2, l)^(l)d 3 x l( f 3 x,. 



(5) 



A quantum mechanical system is described equally well 
by specifying the function K, or by specifying the 
Hamiltonian H from which it results. For some purposes 
the specification in terms of K is easier to use and 
visualize. We desire eventually to discuss quantum 
electrodynamics from this point of view. 

To gain a greater familiarity with the K function and 
the point of view it suggests, we consider a simple 
perturbation problem. Imagine we have a particle in 
a weak potential U(x, t), a function of position and 
time. We wish to calculate K{2, 1) if U differs from 
zero only for t between /1 and t 2 . We shall expand K in 
increasing powers of U: 

K{2, l) = K (2, l)+if (1) (2, l)+if< 2 >(2, 1)+ • • -. (6) 

To zero order in U, K is that for a free particle, K (2, l). 4 
To study the first order correction if (1) (2, 1), first con- 
sider the case that U differs from zero only for the 
infinitesimal time interval Ah between some time h 
and/3+A/ 3 (^i<^3<^). Then if ^(1) is the wave function 
at xi, h, the wave function at x 3 , t z is 



:/ 



*(3)= A'o(3, \)+{l)dH u 



(7) 



since from ti to / 3 the particle is free. For the short 
interval A/ 3 we solve (1) as 

^(x, /■H-A/ 3 ) = exp(-;#A/ 3 )^(x, / 3 ) 

= (l-iH Ah-iUAhU(x, / 3 ), 

4 For a non-relativistic free particle, where 4>„ = exp(/p-x), 
E n = p 2 /2m, (3) gives, as is well known 

K (2, l)=Jexp[-(Jp-x l -»p-x 2 )-ip 2 (/ 2 -^)/2w]^P(2T) -3 
= (27rim- 1 ('2-ii)) _§ exp(^'w(x 2 -x,) 2 (^-/ 1 )- 1 ) 
for h>h, and iT = for t t <h. 



169 



THEORY OF POSITRONS 



751 



where we put H = H -\-U, H being the Hamiltonian 
of a free particle. Thus i^(x, /3+A/3) differs from 
what it would be if the potential were zero (namely 
(l — iH Ah)\p(x, tz)) by the extra piece 

At = - iU(x h h) ■ $(x z , h) A/ 3 , (8) 

which we shall call the amplitude scattered by the 
potential. The wave function at 2 is given by 



i£(x 2 , k) ■■ 



■ I #o(x 2) 



h; x 3 , t 3 +M z )f(xz, h+Ah)d 3 x 3 , 



since after /3+A/3 the particle is again free. Therefore 
the change in the wave function at 2 brought about by 
the potential is (substitute (7) into (8) and (8) into 
the equation for iA(x 2 , / 2 )): 

A^(2) = — i Jk (2, 3)U(3)K (3, 1)^(1)^x^X3^3. 

In the case that the potential exists for an extended 
time, it may be looked upon as a sum of effects from 
each interval A/ 3 so that the total effect is obtained by 
integrating over / 3 as well as x 3 . From the definition (2) 
of K then, we find 



KU(2, l)=-i 



Jk (2, 



3)U(3)K (3,l)dr 3 , (9) 



where the integral can now be extended over all space 
and time, rfr 3 = d 3 x 3 d(z. Automatically there will be no 
contribution if / 3 is outside the range h to t 2 because of 
our definition, K (2, 1) = for h<h- 

We can understand the result (6), (9) this way. We 
can imagine that a particle travels as a free particle 
from point to point, but is scattered by the potential U. 
Thus the total amplitude for arrival at 2 from 1 can 
be considered as the sum of the amplitudes for various 
alternative routes. It may go directly from 1 to 2 
(amplitude K (2, 1), giving the zero order term in (6)). 
Or (see Fig. 1(a)) it may go from 1 to 3 (amplitude 
K (3, 1)), get scattered there by the potential (scatter- 
ing amplitude —iU(3) per unit volume and time) and 
then go from 3 to 2 (amplitude K (2, 3)). This may 
occur for any point 3 so that summing over these 
alternatives gives (9). 

Again, it may be scattered twice by the potential 
(Fig. 1(b)). It goes from 1 to 3 (K (3, 1)), gets scattered 
there ( — iU(3)) then proceeds to some other point, 4, 
in space time (amplitude K (4, 3)) is scattered again 
(— iU(4)) and then proceeds to 2 (K (2, 4)). Summing 
over all possible places and times for 3, 4 find that the 
second order contribution to the total amplitude 
KW(2, 1) is 

(-i) 2 ffK (2,4)U(4)K (4,3) 

XU(3)K (3,l)dT,dT i . (10) 
This can be readily verified directly from (1) just as (9) 





(a) FIRST 0RDER,EQ(9) (b) SECOND ORDER. EQ.(IO) 

Fig. 1. The Schrodinger (and Dirac) equation can be visualized 
as describing the fact that plane waves are scattered successively 
by a potential. Figure 1 (a) illustrates the situation in first order. 
K (2, 3) is the amplitude for a free particle starting at point 3 
to arrive at 2. The shaded region indicates the presence of the 
potential A which scatters at 3 with amplitude — iA(3) per 
cm 3 sec. (Eq. (9)). In (b) is illustrated the second order process 
(Eq. (10)), the waves scattered at 3 are scattered again at 4. How- 
ever, in Dirac one-electron theory iC (4, 3) would represent elec- 
trons both of positive and of negative energies proceeding from 
3 to 4. This is remedied by choosing a different scattering kernel 
iP + (4, 3), Fig. 2. 

was. One can in this way obviously write down any of 
the terms of the expansion (6). 5 

3. TREATMENT OF THE DIRAC EQUATION 

We shall now extend the method of the last section 
to apply to the Dirac equation. All that would seem 
to be necessary in the previous equations is to consider 
H as the Dirac Hamiltonian, ^ as a symbol with four 
indices (for each particle). Then K can still be defined 
by (3) or (4) and is now a 4-4 matrix which operating 
on the initial wave function, gives the final wave func- 
tion. In (10), U(3) can be generalized to A 4 (3) — a-A(3) 
where A 4, A are the scalar and vector potential (times e, 
the electron charge) and a are Dirac matrices. 

To discuss this we shall define a convenient rela- 
tivistic notation. We represent four-vectors like x, t by 
a symbol x„, where \i= 1, 2, 3, 4 and x t = t is real. Thus 
the vector and scalar potential (times e) A, A t is A?. 
The four matrices /3a, /3 can be considered as transform- 
ing as a four vector 7^ (our y m differs from Pauli's by a 
factor i for n= 1, 2, 3). We use the summation conven- 
tion a li b^ = a i b i — a-ibi— ajbi— a 3 6 3 =a-6. In particular if 
a M is any four vector (but not a matrix) we write 
a = a li y li so that a is a matrix associated with a vector 
(a will often be used in place of <z M as a symbol for the 
vector). The 7„ satisfy 7 M 7,.+ 7,7„= 25„„ where 5 4 4= + 1, 
5n= 5 2 2= 5 3 3= — 1, and the other 5„„ are zero. As a 
consequence of our summation convention b liV a,=a ll . 
and 5^=4. Note that ab+ba=2a-b and that a^^a^a,, 
— a- a is a pure number. The symbol d/dx? will mean 
d/dt for ii = 4, and -d/dx, -d/dy, -d/dz for M =l, 
2, 3. Call V=y fi d/dx ll =pd/dt+l3a-V. We shall imagine 

6 We are simply solving by successive approximations an integral 
equation (deducible directly from (1) with H = H +U and (4) 
with H = H ), 

U2)=-ifK (2, 3)U(3)+(3)dr 3 +fKo(2, 1)^(1)^,, 

where the first integral extends over all space and all times *3 
greater than the ti appearing in the second term, and h>h. 



170 



752 



R. P. FE YNM AN 



K (3,3),POS. E. 




JNTAtoS ONLY 
NEG.E 

(b) VIRTUAL SCATTERING (c) VIRTUAL PAIR 
«4>»3 t 4 <t 3 

SECOND ORDER, EO. (14) 

Fig. 2. The Dirac equation permits another solution K + {2, 1) 
if one considers that waves scattered by the potential can proceed 
backwards in time as in Fig. 2 (a). This is interpreted in the second 
order processes (b), (c), by noting that there is now the possi- 
bility (c) of virtual pair production at 4, the positron going to 3 
to be annihilated. This can be pictured as similar to ordinary 
scattering (b) except that the electron is scattered backwards in 
time from 3 to 4. The waves scattered from 3 to 2' in (a) represent 
the possibility of a positron arriving at 3 from 2' and annihilating 
the electron from 1. This view is proved equivalent to hole theory: 
electrons traveling backwards in time are recognized as positrons. 

hereafter, purely for relativistic convenience, that <j> n * 
in (3) is replaced by its adjoint 4> n =4> n *fi- 

Thus the Dirac equation for a particle, mass m, in an 
external field A = A ll y lt is 

(iV-m)il/=A$, (11) 

and Eq. (4) determining the propagation of a free 
particle becomes 

(iV 2 -m)K + (2,l) = id(2,l), (12) 

the index 2 on V 2 indicating differentiation with respect 
to the coordinates x 2li which are represented as 2 in 
K+(2, 1) and 5(2, 1). 

The function K+(2, 1) is defined in the absence of a 
field. If a potential A is acting a similar function, say 
K+ (A) {2, 1) can be defined. It differs from K+(2, 1) by a 
first order correction given by the analogue of (9) 
namely 



#+ ci) (2, i: 



ij K+ (2, 



3)A(3)K+(3, l)dn, (13) 



representing the amplitude to go from 1 to 3 as a free 
particle, get scattered there by the potential (now the 
matrix A(3) instead of U{3)) and continue to 2 as free. 
The second order correction, analogous to (10) is 

# + < 2 >(2,l)=-JJV + (2,4)>i(4) 

XA%(4, 3)A(3)K + (3, l)d u d u , (14) 

and so on. In general K+S A) satisfies 

{iV 2 -A(2)-m)K + ^{2, l) = id(2, 1), (15) 

and the successive terms (13), (14) are the power series 



expansion of the integral equation 

K + ^(2,l) = K + (2,l) 

-i(K + {2, 3)A(3)K + < a \3, l)dr 3 , 



(16) 



which it also satisfies. 

We would now expect to choose, for the special solu- 
tion of (12), K+ = K where #(,(2, 1) vanishes for t 2 <ti 
and for t 2 >h is given by (3) where #„ and E n are the 
eigenfunctions and energy values of a particle satis- 
fying Dirac's equation, and </>„* is replaced by #„. 

The formulas arising from this choice, however, suffer 
from the drawback that they apply to the one electron 
theory of Dirac rather than to the hole theory of the 
positron. For example, consider as in Fig. 1(a) an 
electron after being scattered by a potential in a small 
region 3 of space time. The one electron theory says 
(as does (3) with K + = K ) that the scattered amplitude 
at another point 2 will proceed toward positive times 
with both positive and negative energies, that is with 
both positive and negative rates of change of phase. No 
wave is scattered to times previous to the time of 
scattering. These are just the properties of K (2, 3). 

On the other hand, according to the positron theory 
negative energy states are not available to the electron 
after the scattering. Therefore the choice K + = Ko is 
unsatisfactory. But there are other solutions of (12). 
We shall choose the solution defining K+(2, 1) so that 
K + {2, 1) for h>h is the sum of (3) over positive energy 
states only. Now this new solution must satisfy (12) for 
all times in order that the representation be complete. 
It must therefore differ from the old solution K by a 
solution of the homogeneous Dirac equation. It is clear 
from the definition that the difference K — K + is the 
sum of (3) over all negative energy states, as long as 
t 2 >h. But this difference must be a solution of the 
homogeneous Dirac equation for all times and must 
therefore be represented by the same sum over negative 
energy states also for t 2 <h. Since ^0 = in this case, 
it follows that our new kernel, K+(2, l),for t 2 <h is the 
negative of the sum (J) over negative energy states. That is, 

K + (2,l) = 2Zpos En <l>n(2)tn(l) 

Xexp(-iE n (t 2 -h)) for t 2 >h 

(17) 



-2ZNEGE n <t>n(2)$ n (l) 

Xexp(-iE n (l 2 -ti)) 



for t 2 <h 



With this choice of K + our equations such as (13) and 
(14) will now give results equivalent to those of the 
positron hole theory. 

That (14), for example, is the correct second order 
expression for finding at 2 an electron originally at 1 
according to the positron theory may be seen as follows 
(Fig. 2). Assume as a special example that t 2 >h and 
that the potential vanishes except in interval t 2 —ti so 
that U and / 3 both lie between /i and t 2 . 

First suppose l 4 >t 3 (Fig. 2(b)). Then (since / 3 >/i) 



171 



THEORY OF POSITRONS 



753 



the electron assumed originally in a positive energy 
state propagates in that state (by K+(3, 1)) to position 
3 where it gets scattered (.4(3)). It then proceeds to 4, 
which it must do as a positive energy electron. This is 
correctly described by (14) for j£+(4, 3) contains only 
positive energy components in its expansion, as ti>h. 
After being scattered at 4 it then proceeds on to 2, 
again necessarily in a positive energy state, as t 2 >U. 

In positron theory there is an additional contribution 
due to the possibility of virtual pair production (Fig. 
2(c)). A pair could be created by the potential 4(4) 
at 4, the electron of which is that found later at 2. The 
positron (or rather, the hole) proceeds to 3 where it 
annihilates the electron which has arrived there from 1. 

This alternative is already included in (14) as con- 
tributions for which ti<h, and its study will lead us to 
an interpretation of K + (A, 3) for ti<h. The factor 
K + (2, 4) describes the electron (after the pair produc- 
tion at 4) proceeding from 4 to 2. Likewise K + (3, 1) 
represents the electron proceeding from 1 to 3. K + (A, 3) 
must therefore represent the propagation of the positron 
or hole from 4 to 3. That it does so is clear. The fact 
that in hole theory the hole proceeds in the manner of 
and electron of negative energy is reflected in the fact 
that K+(4, 3) for h<h is (minus) the sum of only 
negative energy components. In hole theory the real 
energy of these intermediate states is, of course, 
positive. This is true here too, since in the phases 
exp(— iE n {h— h)) defining K+(4, 3) in (17), En is nega- 
tive but so is t i — h. That is, the contributions vary with 
t 3 as exp(— i\E n \(h— h)) as they would if the energy 
of the intermediate state were \E n \. The fact that the 
entire sum is taken as negative in computing if+(4, 3) 
is reflected in the fact that in hole theory the amplitude 
has its sign reversed in accordance with the Pauli 
principle and the fact that the electron arriving at 2 
has been exchanged with one in the sea. 6 To this, and 
to higher orders, all processes involving virtual pairs 
are correctly described in this way. 

The expressions such as (14) can still be described as 
a passage of the electron from 1 to 3 (K+(3, 1)), scatter- 
ing at 3 by 4(3), proceeding to 4 (i^+(4, 3)), scattering 
again, 4(4), arriving finally at 2. The scatterings may, 
however, be toward both future and past times, an 
electron propagating backwards in time being recog- 
nized as a positron. 

This therefore suggests that negative energy com- 
ponents created by scattering in a potential be con- 
sidered as waves propagating from the scattering point 
toward the past, and that such waves represent the 
propagation of a positron annihilating the electron in 
the potential. 7 



6 It has often been noted that the one-electron theory apparently 
gives the same matrix elements for this process as does hole theory. 
The problem is one of interpretation, especially in a way that will 
also give correct results for other processes, e.g., self-energy. 

7 The idea that positrons can be represented as electrons with 
proper time reversed relative to true time has been discussed by 
the author and others, particularly by Stuckelberg. E. C. C. 



With this interpretation real pair production is also 
described correctly (see Fig. 3). For example in (13) if 
h<h<h the equation gives the amplitude that if at 
time /i one electron is present at 1, then at time / 2 just 
one electron will be present (having been scattered at 3) 
and it will be at 2. On the other hand if h is less than h, 
for example, if h—t\<tz, the same expression gives the 
amplitude that a pair, electron at 1, positron at 2 will 
annihilate at 3, and subsequently no particles will be 
present. Likewise if h and t x exceed k we have (minus) 
the amplitude for finding a single pair, electron at 2, 
positron at 1 created by 4(3) from a vacuum. If 
h>h>h, (13) describes the scattering of a positron. 
All these amplitudes are relative to the amplitude that 
a vacuum will remain a vacuum, which is taken as 
unity. (This will be discussed more fully later.) 

The analogue of (2) can be easily worked out. 8 It is, 



*(2) 



J- 



(2, \)N{\)*(\)dW 



(18) 



where dWi is the volume element of the closed 3- 
dimensional surface of a region of space time containing 







Fig. 3. Several different processes can be described by the same 
formula depending on the time relations of the variables tz, h. 
Thus P v \K + ^{2, 1)| 2 is the probability that: (a) An electron at 
1 will be scattered at 2 (and no other pairs form in vacuum). 

(b) Electron at 1 and positron at 2 annihilate leaving nothing. 

(c) A single pair at 1 and 2 is created from vacuum, (d) A positron 
at 2 is scattered to 1. (K+. u >(2, 1) is the sum of the effects of 
scattering in the potential to all orders. P v is a normalizing 
constant.) 



Stuckelberg, Helv. Phys. Acta 15, 23 (1942); R. P. Feynman, 
Phys. Rev. 74, 939 (1948). The fact that classically the action 
(proper time) increases continuously as one follows a trajectory 
is reflected in quantum mechanics in the fact that the phase, which 
is \E n \ \h—li\, always increases as the particle proceeds from one 
scattering point to the next. 

8 By multiplying (12) on the right by (— tVi— m) and noting 
that Vi«(2, 1) = -Vj5(2, 1) show that .£+(2,1) also satisfies 
K+(2, 1)(— iV\— w)=z'5(2, 1), where the Vi operates on variable 
1 in K+(2, 1) but is written after that function to keep the correct 
order of the y matrices. Multiply this equation by ^(1) and Eq. 
(11) (with 4 = 0, calling the variables 1) by K+(2, 1), subtract 
and integrate over a region of space-time. The integral on the left- 
hand side can be transformed to an integral over the surface of 
the region. The right-hand side is \p(2) if the point 2 lies within 
the region, and is zero otherwise. (What happens when the 3- 
surface contains a light line and hence has no unique normal need 
not concern us as these points can be made to occur so far away 
from 2 that their contribution vanishes.) 



172 



754 



FE YN M AN 



point 2, and N(l) is ^(1)7,, where A^ M (1) is the inward 
drawn unit normal to the surface at the point 1. That 
is, the wave function ^(2) (in this case for a free par- 
ticle) is determined at any point inside a four-dimen- 
sional region if its values on the surface of that region 
are specified. 

To interpret this, consider the case that the 3-surface 
consists essentially of all space at some time say t = 
previous to t 2 , and of all space at the time T>t 2 . The 
cylinder connecting these to complete the closure of the 
surface may be very distant from x 2 so that it gives no 
appreciable contribution (as K + (2, 1) decreases expo- 
nentially in space-like directions). Hence, if 74 = /?, since 
the inward drawn normals TV" will be /? and — /3, 



*(2): 



jV 



(2, i)W{i)ffixi 



/ 



K + (2, VW(l')(Pxv, (19) 



where /i = 0, ty — T. Only positive energy (electron) 
components in ^(1) contribute to the first integral and 
only negative energy (positron) components of ^(1') to 
the second. That is, the amplitude for finding a charge 
at 2 is determined both by the amplitude for finding 
an electron previous to the measurement and by the 
amplitude for finding a positron after the measurement. 
This might be interpreted as meaning that even in a 
problem involving but one charge the amplitude for 
finding the charge at 2 is not determined when the only 
thing known in the amplitude for finding an electron 
(or a positron) at an earlier time. There may have been 
no electron present initially but a pair was created in 
the measurement (or also by other external fields). The 
amplitude for this contingency is specified by the 
amplitude for finding a positron in the future. 

We can also obtain expressions for transition ampli- 
tudes, like (5). For example if at / = we have an elec- 
tron present in a state with (positive energy) wave 
function /(x), what is the amplitude for finding it at 
t=T with the (positive energy) wave function g(x)? 
The amplitude for finding the electron anywhere after 
/ = is given by (19) with ^(1) replaced by /(x), the 
second integral vanishing. Hence, the transition ele- 
ment to find it in state g(x) is, in analogy to (5), just 
(t 2 =T,t 1 = 0) 



I 



g{x 2 )&K + {2, DjS/CxOAidhc, 



(20) 



since g* = <?/3. 

If a potential acts somewhere in the interval between 
and T, K+ is replaced by K+ (A) . Thus the first order 
effect on the transition amplitude is, from (13), 

-if g(x 2 )0K + (2, 3)A(3)K + (3, l)/3/(x 1 )rf 3 x 1 d 3 x 2 . (21) 

Expressions such as this can be simplified and the 
3-surface integrals, which are inconvenient for rela- 



tivistic calculations, can be removed as follows. Instead 
of defining a state by the wave function /(x), which it 
has at a given time /i = 0, we define the state by the 
function /(l) of four variables Xi, h which is a solution 
of the free particle equation for all h and is /(xi) for 
/i = 0. The final state is likewise defined by a function 
g(2) over-all space-time. Then our surface integrals can 
be performed since fK + (3, l) i 8/(xi)d 3 x 1 =/(3) and 
fg(x 2 )P(Px 2 K + (2, 3) = 0(3). There results 



ifg(3)A(3)f(3)di 



(22) 



the integral now being over-all space-time. The transi- 
tion amplitude to second order (from (14)) is 



-IS 



g(2)A(2)K + (2, l)A{\)f{\)dT,dr 2 , (23) 



for the particle arriving at 1 with amplitude /(l) is 
scattered (.4(1)), progresses to 2, (K + (2, 1)), and is 
scattered again (.4(2)), and we then ask for the ampli- 
tude that it is in state g(2). If g(2) is a negative energy 
state we are solving a problem of annihilation of elec- 
tron in /(l), positron in g(2), etc. 

We have been emphasizing scattering problems, but 
obviously the motion in a fixed potential V, say in a 
hydrogen atom, can also be dealt with. If it is first 
viewed as a scattering problem we can ask for the 
amplitude, 0*(1), that an electron with original free 
wave function was scattered k times in the potential V 
either forward or backward in time to arrive at 1. Then 
the amplitude after one more scattering is 



**+i(2) 



= -iJK42, 



DVmMVdn. 



(24) 



An equation for the total amplitude 

*(i)«£ **(i) 

ft=0 

for arriving at 1 either directly or after any number of 
scatterings is obtained by summing (24) over all k from 
Oto 00; 



lK2) = o (2) 



-•/* 



(2,1)7(1)^(1)^7 



(25) 



Viewed as a steady state problem we may wish, for 
example, to find that initial condition 4> (or better just 
the \p) which leads to a periodic motion of \f/. This is 
most practically done, of course, by solving the Dirac 
equation, 

(tV-»)*(l)«K(l)*(l), (26) 

deduced from (25) by operating on both sides by iV 2 —m, 
thereby eliminating the </> , and using (12). This illus- 
trates the relation between the points of view. 

For many problems the total potential A-\- V may be 
split conveniently into a fixed one, V, and another, A, 
considered as a perturbation. If A' + (n is defined as in 



173 



THEORY OF POSITRONS 



755 



(16) with V for A, expressions such as (23) are valid 
and useful with K+ replaced by K+ (V) and the functions 
/(l), g(2) replaced by solutions for all space and time 
of the Dirac Eq. (26) in the potential V (rather than 
free particle wave functions). 

4. PROBLEMS INVOLVING SEVERAL CHARGES 

We wish next to consider the case that there are two 
(or more) distinct charges (in addition to pairs they may 
produce in virtual states). In a succeeding paper we 
discuss the interaction between such charges. Here we 
assume that they do not interact. In this case each 
particle behaves independently of the other. We can 
expect that if we have two particles a and b, the ampli- 
tude that particle a goes from Xi at h, to x 3 at t 3 while 
b goes from x 2 at t% to x 4 at U is the product 

K{3, 4; 1, 2) = K +a (3, l)K +b (4, 2). 

The symbols a, b simply indicate that the matrices 
appearing in the K + apply to the Dirac four component 
spinors corresponding to particle a or b respectively (the 
wave function now having 16 indices). In a potential 
K +a and K +b become K + J A) and K +b ^ where K +a ^ A > 
is defined and calculated as for a single particle. They 
commute. Hereafter the a, b can be omitted; the space 
time variable appearing in the kernels suffice to define 
on what they operate. 

The particles are identical however and satisfy the 
exclusion principle. The principle requires only that one 
calculate K{3, 4; 1, 2)-X(4, 3; 1, 2) to get the net 
amplitude for arrival of charges at 3, 4. (It is normalized 
assuming that when an integral is performed over points 
3 and 4, for example, since the electrons represented are 
identical, one divides by 2.) This expression is correct 
for positrons also (Fig. 4). For example the amplitude 
that an electron and a positron found initially at x x and 
x 4 (say ti = ti) are later found at x 3 and x 2 (with 
t2 = tz>h) is given by the same expression 

K + ^(3, l)K + ^(4, 2)-# + ">(4, l)K + ^(3, 2). (27) 

The first term represents the amplitude that the electron 
proceeds from 1 to 3 and the positron from 4 to 2 (Fig. 
4(c)), while the second term represents the interfering 
amplitude that the pair at 1, 4 annihilate and what is 
found at 3, 2 is a pair newly created in the potential. 
The generalization to several particles is clear. There is 
an additional factor K + U) for each particle, and anti- 
symmetric combinations are always taken. 

No account need be taken of the exclusion principle 
in intermediate states. As an example consider again 
expression (14) for h>h and suppose U<h so that the 
situation represented (Fig. 2(c)) is that a pair is made 
at 4 with the electron proceeding to 2, and the positron 
to 3 where it annihilates the electron arriving from 1. 
It may be objected that if it happens that the electron 
created at 4 is in the same state as the one coming from 
1, then the process cannot occur because of the exclusion 
principle and we should not have included it in our 








(c) 

Fig. 4. Some problems involving two distinct charges (in addi- 
tion to virtual pairs they may produce) : P v \K + ( - A, (3, \)K + (A) (4, 2) 
-#+ u >(4, 1)JC + M)(3, 2) | 2 is the probability that: (a) Electrons 
at 1 and 2 are scattered to 3, 4 (and no pairs are formed), (b) 
Starting with an electron at 1 a single pair is formed, positron at 2, 
electrons at 3, 4. (c) A pair at 1, 4 is found at 3, 2, etc. The exclu- 
sion principle requires that the amplitudes for processes involving 
exchange of two electrons be subtracted. 

term (14). We shall see, however, that considering the 
exclusion principle also requires another change which 
reinstates the quantity. 

For we are computing amplitudes relative to the 
amplitude that a vacuum at h will still be a vacuum at 
t 2 . We are interested in the alteration in this amplitude 
due to the presence of an electron at 1. Now one process 
that can be visualized as occurring in the vacuum is the 
creation of a pair at 4 followed by a re-annihilation of 
the same pair at 3 (a process which we shall call a closed 
loop path). But if a real electron is present in a certain 
state 1, those pairs for which the electron was created 
in state 1 in the vacuum must now be excluded. We 
must therefore subtract from our relative amplitude the 
term corresponding to this process. But this just rein- 
states the quantity which it was argued should not 
have been included in (14), the necessary minus sign 
coming automatically from the definition of K + . It is 
obviously simpler to disregard the exclusion principle 
completely in the intermediate states. 

All the amplitudes are relative and their squares give 
the relative probabilities of the various phenomena. 
Absolute probabilities result if one multiplies each of 
the probabilities by P v , the true probability that if one 
has no particles present initially there will be none 
finally. This quantity P v can be calculated by normal- 
izing the relative probabilities such that the sum of the 
probabilities of all mutually exclusive alternatives is 
unity. (For example if one starts with a vacuum one can 
calculate the relative probability that there remains a 



174 



756 



R. P. FE YNM AN 



vacuum (unity), or one pair is created, or two pairs, etc. 
The sum is P„ -1 .) Put in this form the theory is com- 
plete and there are no divergence problems. Real proc- 
esses are completely independent of what goes on in 
the vacuum. 

When we come, in the succeeding paper, to deal with 
interactions between charges, however, the situation is 
not so simple. There is the possibility that virtual elec- 
trons in the vacuum may interact electromagnetically 
with the real electrons. For that reason processes occur- 
ing in the vacuum are analyzed in the next section, in 
which an independent method of obtaining P v is 
discussed. 

5. VACUUM PROBLEMS 

An alternative way of obtaining absolute amplitudes 
is to multiply all amplitudes by C v , the vacuum to 
vacuum amplitude, that is, the absolute amplitude that 
there be no particles both initially and finally. We can 
assume C v = 1 if no potential is present during the 
interval, and otherwise we compute it as follows. It 
differs from unity because, for example, a pair could be 
created which eventually annihilates itself again. Such 
a path would appear as a closed loop on a space-time 
diagram. The sum of the amplitudes resulting from all 
such single closed loops we call L. To a first approxima- 
tion L is 



L«: 



-ffsKK + (2,l)A(X) 

XK + (l,2)A(2)yT l dT2. 



(28) 



For a pair could be created say at 1, the electron and 
positron could both go on to 2 and there annihilate. 
The spur, Sp, is taken since one has to sum over all 
possible spins for the pair. The factor \ arises from the 
fact that the same loop could be considered as starting 
at either potential, and the minus sign results since the 
interactors are each —iA. The next order term would be 9 

D» = + (i/3)fffsptK + (2, 1)4(1) 

XK+(1, 3)A(3)K+(3, 2)i4(2)]dT 1 dT 2 rfT 3 , 

etc. The sum of all such terms gives Z,. 10 

9 This term actually vanishes as can be seen as follows. In any 
spur the sign of all y matrices may be reversed. Reversing the 
sign of 7 in K+(2, 1) changes it to the transpose of K + (l, 2) so 
that the order of all factors and variables is reversed. Since the 
integral is taken over all n, r 2 , and t% this has no effect and we are 
left with (— l) 3 from changing the sign of A. Thus the spur equals 
its negative. Loops with an odd number of potential interactors 
give zero. Physically this is because for each loop the electron can 
go around one way or in the opposite direction and we must add 
these amplitudes. But reversing the motion of an electron makes 
it behave like a positive charge thus changing the sign of each 
potential interaction, so that the sum is zero if the number of 
interactions is odd. This theorem is due to W. H. Furry, Phys. 
Rev. 51, 125(1937). 

10 A closed expression for L in terms of K+<- A) is hard to obtain 
because of the factor (1/n) in the nth term. However, the per- 
turbation in L, AL due to a small change in potential AA, is easy 
to express. The (l/«) is canceled by the fact that AA can appear 



In addition to these single loops we have the possi- 
bility that two independent pairs may be created and 
each pair may annihilate itself again. That is, there may 
be formed in the vacuum two closed loops, and the 
contribution in amplitude from this alternative is just 
the product of the contribution from each of the loops 
considered singly. The total contribution from all such 
pairs of loops (it is still consistent to disregard the 
exclusion principle for these virtual states) is L 2 /2 for 
in L 2 we count every pair of loops twice. The total 
vacuum-vacuum amplitude is then 



C v =l-L+L 2 /2-L*/6+ 



ixp(-L), (30) 



the successive terms representing the amplitude from 
zero, one, two, etc., loops. The fact that the contribu- 
tion to C v of single loops is — L is a consequence of the 
Pauli principle. For example, consider a situation in 
which two pairs of particles are created. Then these 
pairs later destroy themselves so that we have two 
loops. The electrons could, at a given time, be inter- 
changed forming a kind of figure eight which is a single 
loop. The fact that the interchange must change the 
sign of the contribution requires that the terms in C v 
appear with alternate signs. (The exclusion principle is 
also responsible in a similar way for the fact that the 
amplitude for a pair creation is —K+ rather than +K + .) 
Symmetrical statistics would lead to 

C„ = 1 + L+ L 2 /2 = exp(+ L) . 

The quantity L has an infinite imaginary part (from 
L m , higher orders are finite). We will discuss this in 
connection with vacuum polarization in the succeeding 
paper. This has no effect on the normalization constant 
for the probability that a vacuum remain vacuum is 
given by 

Pv= |C„| 2 =exp(-2-real part of L), 

from (30). This value agrees with the one calculated 
directly by renormalizing probabilities. The real part 
of L appears to be positive as a consequence of the Dirac 
equation and properties of K + so that P v is less than 
one. Bose statistics gives C = exp(+-£) and conse- 
quently a value of P v greater than unity which appears 
meaningless if the quantities are interpreted as we have 
done here. Our choice of K+ apparently requires the 
exclusion principle. 

Charges obeying the Klein-Gordon equation can be 
equally well treated by the methods which are dis- 
cussed here for the Dirac electrons. How this is done is 
discussed in more detail in the succeeding paper. The 
real part of L comes out negative for this equation so 
that in this case Bose statistics appear to be required 
for consistency. 3 

in any of the n potentials. The result after summing over n by 
(13), (14) and using (16) is 

AL=-ifspl(K^Ki, i)~K + (l, l))Ail(l)]rfTi. (29) 

The term #+(1, 1) actually integrates to zero. 



175 



THEORY OF POSITRONS 



757 



6. ENERGY-MOMENTUM REPRESENTATION 

The practical evaluation of the matrix elements in 
some problems is often simplified by working with 
momentum and energy variables rather than space and 
time. This is because the function K + (2, 1) is fairly 
complicated but we shall find that its Fourier transform 
is very simple, namely (i/ATr 2 ){p—m)- x that is 



K + (2,l) = (i/W) 



/<>- 



i)- 1 exp(-ip-x 2 i)d i p, (31) 



where p-x 2 i=p-x 2 —p-xi=p )l x 2 ^—p ll xi ll , P=p v n ll ., and 
d*p means (2ir)~ 2 dpidp 2 dpzdpi, the integral over all p. 
That this is true can be seen immediately from (12), 
for the representation of the operator iV— m in energy 
(pi) and momentum (pi, 2, 3) space is p— m and the trans- 
form of 5(2, 1) is a constant. The reciprocal matrix 
(p—m)- 1 can be interpreted as (/i+ffl)^-^)" 1 for 
pp—m 2 = {p—m)(p-\-m) is a pure number not involving 
7 matrices. Hence if one wishes one can write 



where 



K + (2,l) = i(iV 2 +m)I + (2,l), 



I + (2, 1) = (2tt)- 2 f(^-»V exp(-ip-x n )d% (32) 

is not a matrix operator but a function satisfying 

□ 2 2 / + (2, \)-m 2 I + (2, 1) = 8(2, 1), (33) 

where -[J 2 2 = (V 2 ) 2 = (d/dx u )(d/dx 2lt ). 

The integrals (31) and (32) are not yet completely 
defined for there are poles in the integrand when 
pp—m 2 =0. We can define how these poles are to be 
evaluated by the rule that m is considered to have an 
infinitesimal negative imaginary part. That is m, is re- 
placed by m— id and the limit taken as 5— >0 from above. 
This can be seen by imagining that we calculate K+ by 
integrating on pi first. If we call E= + (m?+pi 2 
+/ > 2 2 +/>3 2 ) i then the integrals involve pi essentially as 
S exp(— ipi(t 2 — h))dpi(pi 2 — E?)~ l which has poles at 
p t =+E and pi=—E. The replacement of m by m— i8 
means that E has a small negative imaginary part; the 
first pole is below, the second above the real axis. Now 
if t 2 — h>0 the contour can be completed around the 
semicircle below the real axis thus giving a residue from 
the pi=+E pole, or -(2E)~ l exp(-iE(t 2 -ti)). If 
k— h<0 the upper semicircle must be used, and 
Pi=—Ea.t the pole, so that the function varies in each 
case as required by the other definition (17). 

Other solutions of (12) result from other prescrip- 
tions. For example if pi in the factor (p 2 —m 2 )~ l is con- 
sidered to have a positive imaginary part K+ becomes 
replaced by K , the Dirac one-electron kernel, zero for 
t 2 <t\. Explicitly the function is 11 (x, t=x 2 i li ) 

I + (x, t)=- (iir)- 1 8(s 2 )+ (m/Zirs)!!^ (ms), (34) 

where s= + (t 2 -x 2 )* for fi>x 2 and s= -i(x 2 -t 2 )* for 

n I+{x,t) is {2i)-i{Di{x,t)-iD{x,t)) where D, and D are the 
functions defined by W. Pauli, Rev. Mod. Phys. 13, 203 (1941). 



t 2 <x 2 , Hi (2) is the Hankel function and 8(s 2 ) is the 
Dirac delta function of s 2 . It behaves asymptotically 
as exp(— ims), decaying exponentially in space-like 
directions. 12 

By means of such transforms the matrix elements 
like (22), (23) are easily worked out. A free particle 
wave function for an electron of momentum pi is 
U\ exp(— ipi-x) where u\ is a constant spinor satisfying 
the Dirac equation p\U\ = mu\ so that pi 2 = m 2 . The 
matrix element (22) for going from a state p u u x to a 
state of momentum p 2 , spinor u%, is — Air 2 i(u 2 a(q)ui) 
where we have imagined A expanded in a Fourier 
integral 



A(\) 



= I a(q) exp(—iq-xi)d 4 q, 



and we select the component of momentum q=p 2 —p\. 
The second order term (23) is the matrix element 
between U\ and u 2 of 



-Wi((a(p 2 



■pi-q))(J>i+q-m)- l a(q)d*q, (35) 



since the electron of momentum pi may pick up q from 
the potential a(q), propagate with momentum pi+q 
(factor (p\-\-q— w) _1 ) until it is scattered again by the 
potential, a(p 2 —pi—q), picking up the remaining mo- 
mentum, p 2 —pi—q, to bring the total to p 2 . Since all 
values of q are possible, one integrates over q. 

These same matrices apply directly to positron prob- 
lems, for if the time component of, say, pi is negative 
the state represents a positron of four-momentum —pi, 
and we are describing pair production if p 2 is an elec- 
tron, i.e., has positive time component, etc. 

The probability of an event whose matrix element is 
(u 2 Mui) is proportional to the absolute square. This 
may also be written (uiMu 2 )(u 2 Mui), where M is M 
with the operators written in opposite order and explicit 
appearance of i changed to — i(M is /3 times the complex 
conjugate transpose of /3M). For many problems we are 
not concerned about the spin of the final state. Then we 
can sum the probability over the two u 2 corresponding 
to the two spin directions. This is not a complete set be- 
cause p 2 has another eigenvalue, — m. To permit sum- 
ming over all states we can insert the projection operator 
(2m)~ l (p 2 -\-m) and so obtain (2m)~ l (iiiM(p 2 -\-m)Mui) 
for the probability of transition from pi, u u to p 2 with 
arbitrary spin. If the incident state is unpolarized we 
can sum on its spins too, and obtain 



(2 m)~ 2 Sp\_(pi+ m)M(p 2 + m)M ] 



(36) 



for (twice) the probability that an electron of arbitrary 
spin with momentum pi will make transition to p 2 . The 
expressions are all valid for positrons when p's with 

12 If the —id is kept with m here too the function I + approaches 
zero for infinite positive and negative times. This may be useful 
in general analyses in avoiding complications from infinitely 
remote surfaces. 



176 



758 



FEYNMAN 



negative energies are inserted, and the situation inter- 
preted in accordance with the timing relations discussed 
above. (We have used functions normalized to (uu) = 1 
instead of the conventional (uj3u) = (u*u) = 1 . On our 
scale (ufiu) = energy/ w so the probabilities must be 
corrected by the appropriate factors.) 

The author has many people to thank for fruitful 
conversations about this subject, particularly H. A. 
Bethe and F. J. Dyson. 

APPENDIX 

a. Deduction from Second Quantization 

In this section we shall show the equivalence of this theory with 
the hole theory of the positron. 2 According to the theory of second 
quantization of the electron field in a given potential, 13 the state 
of this field at any time is represented by a wave function x 
satisfying 

idx/dt=H X , 
where H= y**(x)(a-(-iV-A)+^ 4 +w/3)*(x)(f 3 x and *(x) is 
an operator annihilating an electron at position x, while **(x) is 
the corresponding creation operator. We contemplate a situation 
in which at t = we have present some electrons in states repre- 
sented by ordinary spinor functions /i(x), /2(x), • ■ ■ assumed 
orthogonal, and some positrons. These are described as holes in 
the negative energy sea, the electrons which would normally fill the 
holes having wave functions pi(x'), pi{x), •••. We ask, at time T 
what is the amplitude that we find electrons in states gi(x), 
gt(x), ■ ■ ■ and holes at qi(x),qz(x), • • •. If the initial and final state 
vectors representing this situation are xi and xi respectively, we 
wish to calculate the matrix element 

R = \xi* exp(-^ r #^)xi)=(x/*Sx.). (37) 

We assume that the potential A differs from zero only for times 
between and T so that a vacuum can be defined at these times. 
If xo represents the vacuum state (that is, all negative energy 
states filled, all positive energies empty), the amplitude for having 
a vacuum at time T, if we had one at / = 0, is 



C^=(xo*5xo), 



(38) 



writing 5 for exp( — if T Hdt). Our problem is to evaluate R and 
show that it is a simple factor times C„, and that the factor involves 
the K + iA > functions in the way discussed in the previous sections. 
To do this we first express x; in terms of xo- The operator 



/ 



**(x)0(x)rf 3 x, 



(39) 



creates an electron with wave function <j>(x). Likewise <f>= f<t>*(x) 
X*(x)<f 3 x annihilates one with wave function 4>{x). Hence state 
x. is xi = Fi*F?*- ■ PiP-2- • -xo while the final state is 6VG 2 *- ' " 
XQ\Qi- ■ -xo where F,-, G„ Pi, Qi are operators defined like <I>, in 
(39), but with/, gi, pi, qi replacing <)>; for the initial state would 
result from the vacuum if we created the electrons in j\, / 2 , 
and annihilated those in pi, pi, ••-. Hence we must find 

R = (xo* • • ■ QSQi* ■ ■ ■ G i G 1 SF 1 *F i * • • • P,P 2 • • • xo) . (40) 
To simplify this we shall have to use commutation relations be- 
tween a <i>* operator and S. To this end consider exp( — iJl l Hdt')$* 
Xexp(+iJl'Hdt') and expand this quantity in terms of **(x), 
giving f**(x)<t>(x,t)d 3 x, (which defines <t>(x,t)). Now multiply 
this equation by exp(+iJl'Hdt') ■ ■ -exp(-iJl'Hdt') and find 

JV(x)tf>(x)<f 3 x = JV(x, l)4>(x, t)d% (41) 

where we have defined ¥(x, t) by V(x, l) = exp(+if 'Hdt')y(x) 

13 See, for example, G. Wentzel, Einfuhrung in die Quanten- 
theorie der Wellenfeldcr (Franz Deuticke, Leipzig, 1943), Chap- 
ter V. 



Xexpi — iJl'Hdt'). As is well known *(x, t) satisfies the Dirac 
equation, (differentiate *(x, t) with respect to t and use commuta- 
tion relations of H and *) 

id¥(x, 0/d<=(a-(-*V-A)-Md-*»/3)*(x, I). (42) 

Consequently </>(x, t) must also satisfy, the Dirac equation (differ- 
entiate (41) with respect to t, use (42) and integrate by parts). 

That is, if <t>(x, T) is that solution of the Dirac equation at time 
T which is <t>(x) at t=0, and if we define ** = y**(x)^(x)d 3 x and 
*'* = y**(x)<>(x, T)d 3 x then *'* = 5**5 _1 , or 



S** = <*>'*£. 



(43) 



The principle on which the proof will be based can now be 
illustrated by a simple example. Suppose we have just one electron 
initially and finally and ask for 



■■(xo*GSF*xo)- 



(44) 



We might try putting F* through the operator 5 using (43), 
SF* = F'*S, where/' in F'* : = y**(x)/'(x)<f 3 x is the wave function 
at T arising from /(x) at 0. Then 

r=(xo*GF'*Sx a )=fg*(x)f'(x)d*x-C„-(xo*F'*GS X o), (45) 

where the second expression has been obtained by use of the defi- 
nition (38) of C v and the general commutation relation 



GF*+F*G = fg*(x)f(x)d% 



which is a consequence of the properties of *(x) (the others are 
FG= —GF and F*G*=—G*F*). Now xo*F'* in the last term in 
(45) is the complex conjugate of F'xo- Thus if/' contained only 
positive energy components, f'xo would vanish and we would have 
reduced r to a factor times C v . But F', as worked out here, does 
contain negative energy components created in the potential A 
and the method must be slightly modified. 

Before putting F* through the operator we shall add to it 
another operator F"* arising from a function f"(x) containing only 
negative energy components and so chosen that the resulting /' 
has only positive ones. That is we want 

S(F poe *+F aeg "*) = F poa '*S, (46) 

where the "pos" and "neg" serve as reminders of the sign of the 
energy components contained in the operators. This we can now 
use in the form 

SF poB * = F poa '*S-SFne g "*. (47) 

In our one electron problem this substitution replaces r by two 
terms 



,= (xo*C7F po8 '*Sxo)-(xo*GSF 
The first of these reduces to 



*xo). 



' = JV«/poa'(x)d 3 X-C„ 



as above, for Fp OS 'xo is now zero, while the second is zero since the 
creation operator F aes "* gives zero when acting on the vacuum 
state as all negative energies are full. This is the central idea of 
the demonstration. 

The problem presented by (46) is this: Given a function / pos (x) 
at time 0, to find the amount, /neg", of negative energy component 
which must be added in order that the solution of Dirac's equa- 
tion at time T will have only positive energy components, / po3 '. 
This is a boundary value problem for which the kernel K + U) is 
designed. We know the positive energy components initially, / pos , 
and the negative ones finally (zero). The positive ones finally are 
therefore (using (19)) 

/po.'(x*)=/jr + W>(2, l)/3/ po9 (x 1 )d 3 x 1 , (48) 

where U_= T, t x = Q. Similarly, the negative ones initially are 

/neg"(x 2 ) =/aV- 4) (2, l)/3/ P os(Xl)<f 3 X 1 -/ P os(x 2 ), (49) 

where h approaches zero from above, and ii = 0. The / pos (x>) is 



177 



THEORY OF POSITRONS 



759 



subtracted to keep in / ne g"(x 2 ) only those waves which return 
from the potential and not those arriving directly at t 2 from the 
K + (2, 1) part of AV X) (2, 1), as lz— -0. We could also have written 



The value of Cv(k— At ) arises from the Hamiltonian II to— Ma 
which differs from Hi just by having an extra potential during 
the short interval A/ - Hence, to first order in Al , we have 



Therefore the one-electron problem 
gives by (48) 



r = fg*(x)f poa '(x)d 3 x-C„, 



/«.«"<**) =/lX+ W) (2, l)-A\(2,l)]/3/ po8 (x I )^x 1 . (50) CMo-Uo) = (xo*exp(-if t T _ M 7/« -a^)xo) 

= (xo* exp(- if* Htodt)\l-iM f**(x) 

X(-a-A(x,<o)+^ 4 (x,/ ))*(x^ 3 xJxo); 
we therefore obtain for the derivative of Cv the expression 

-dC c (lo)/dt = -i(xo* exp(-if T HtodtJ 

xJV(x)/3/l(x,/o)*(x)rf 3 xxo), (51) 



■■ Cvfg*(x 2 )K+^(2, VM*i)d 3 Xid 3 x 2 , 



as expected in accordance with the reasoning of the previous sec- 
tions (i.e., (20) with AV X) replacing A' + ). 

The proof is readily extended to the more general expression R, 
(40), which can be analyzed by induction. First one replaces Pi* 
by a relation such as (47) obtaining two terms 

R=(xo*- ■ -Qi*Qi*- ■ -GzG^po^SF,*-- -P X P 2 - • - X o) 

- (xo* • • ■ Q2*Qi* ■ ■ ■ G 2 G l SF laeg "*F 2 * ■ ■ ■ PJ 2 • • • xo) ■ 
In the first term the order of Pi pos '* and Gi is then interchanged, 
producing an additional term ygi*(x)/i P0 s'(x)<f 3 x times an expres- 
sion with one less electron in initial and final state. Next it is 
exchanged with G 2 producing an addition — J % g 2 *(x)fi poa '(x)d i x 
times a similar term, etc. Finally on reaching the Q* with which 
it anticommutes it can be simply moved over to juxtaposition 
with xo* where it gives zero. The second term is similarly handled 
by moving Fmeg"* through anti commuting F 2 *, etc., until it 
reaches Pi. Then it is exchanged with P x to produce an addi- 
tional simpler term with a factor = Fj , pi*(x)fi ueli "(x)d 3 x or 
z ¥fpi*(x2)K+^(2, lWKxO^XKPxjfrom (49), with ^ = ^ = (the 
extra /i(x 2 ) in (49) gives zero as it is orthogonal to pi(x 2 )). This 
describes in the expected manner the annihilation of the pair, 
electron /i, positron pi. The F aeg "* is moved in this way succes- 
sively through the P's until it gives zero when acting on xo- Thus 
R is reduced, with the expected factors (and with alternating signs 
as required by the exclusion principle), to simpler terms containing 
two less operators which may in turn be further reduced by using 
P 2 * in a similar manner, etc. After all the P* are used the Q*'s 
can be reduced in a similar manner. They are moved through the 
5 in the opposite direction in such a manner as to produce a purely 
negative energy operator at time 0, using relations analogous to 
(46) to (49). After all this is done we are left simply with the ex- 
pected factor times C» (assuming the net charge is the same in 
initial and final state.) 

In this way we have written the solution to the general problem 
of the motion of electrons in given potentials. The factor Cv is 
obtained by normalization. However for photon fields it is desir- 
able to have an explicit form for Cv in terms of the potentials. 
This is given by (30) and (29) and it is readily demonstrated that 
this also is correct according to second quantization. 

b. Analysis of the Vacuum Problem 

We shall calculate Cv from second quantization by induction 
considering a series of problems each containing a potential dis- 
tribution more nearly like the one we wish. Suppose we know C v 
for a problem like the one we want and having the same potentials 
for time t between some to and T, but having potential zero for 
times from to to. Call this Cvito), the corresponding Hamiltonian 
Hto and the sum of contributions for all single loops, L(t a ). Then 
for to= T we have zero potential at all times, no pairs can be 
produced, L(T) = and Cv(T) = l. For t a =0 we have the com- 
plete problem, so that C v (0) is what is defined as Cv in (38). 
Generally we have, 



:*('o) = (xo* exp(-^ T Htodtjy 



= {xo*expy-if t Htodtjxoj, 



since Hto is identical to the constant vacuum Hamiltonian Ht for 
Kto and xo is an eigenfunction of Ht with an eigenvalue (energy 
of vacuum) which we can take as zero. 



which will be reduced to a simple factor times Cv{t a ) by methods 
analogous to those used in reducing R. The operator * can be 
imagined to be split into two pieces ^pos and ^oeg operating on 
positive and negative energy states respectively. The * pos on xo 
gives zero so we are left with two terms in the current density, 
■tpos^^neg and ¥ ae g*0Ay Dee . The latter ^neg^^vE'neg is just 
•the expectation value of (5A taken over all negative energy states 
(minus ^neg/^^neg* which gives zero acting on xo)- This is the 
effect of the vacuum expectation current of the electrons in the 
sea which we should have subtracted from our original Hamil- 
tonian in the customary way. 

The remaining term ^ p03 */3^4^neg, or its equivalent SE f pos */3i4*' 
can be considered as **(x)f p03 (x) where f pos (x) is written for the 
positive energy component of the operator QA^ (x). Now this 
operator, *'*(x)f pos (x), or more precisely just the **(x) part of it, 
can be pushed through the exp( — iJlo T Hdl) in a manner exactly 
analogous to (47) when / is a function. (An alternative derivation 
results from the consideration that the operator ^(x, t) which 
satisfies the Dirac equation also satisfies the linear integral equa- 
tions which are equivalent to it.) That is, (51) can be written 
by (48), (50), 

-dCv(to)/dt = -i(xo*ff**(x2)K + ^(2, 1) 

Xexp(-*J^ fftf^aMxO^x^Xaxo) 

+ *(xo* exp(-ij^ Hdt)ff**(x 2 )lK + ^(2, 1) 

-K + (2, l)2A(l)^(x 1 )d 3 x 1 d 3 x 2X o) i 

where in the first term t 2 =T, and in the second t 2 -*t =ti. The 
(A) in K + (A) refers to that part of the potential A after t . The 
first term vanishes for it involves (from the K + u) (2, 1)) only 
positive energy components of **, which give zero operating into 
xo*. In the second term only negative components of ^*(x 2 ) 
appear. If, then ^*(x 2 ) is interchanged in order with ^(xi) it will 
give zero operating on xo, and only the term, 

-dC,(to)/dh=+ifspl(K + w(l, 1) 

-K+a, l)M(l)]i 3 xi-C„(/ ), (52) 

will remain, from the usual commutation relation of ^* and ^. 

The factor of Cv(t ) in (52) times -Mo is, according to (29) 
(reference 10), just L(t — At ) — L(t ) since this difference arises 
from the extra potential AA = A during the short time interval 
A* . Hence -dCv(to)/dt = + (dL(to)/dto)Cv(t ) so that integration 
from h= T to t =0 establishes (30). 

Starting from the theory of the electromagnetic field in second 
quantization, a deduction of the equations for quantum electro- 
dynamics which appear in tne succeeding paper may be worked 
out using very similar principles. The Pauli-Weisskopf theory of 
the Klein-Gordon equation can apparently be analyzed in essen- 
tially the same way as that used here for Dirac electrons. 



178 



PHYSICAL REVIEW 



VOLUME 76, NUMBER 6 



SEPTEMBER 



Space-Time Approach to Quantum Electrodynamics 

R. P. Feynman 

Department of Physics, Cornell University, Ithaca, New York 

(Received May 9, 1949) 



In this paper two things are done. (1) It is shown that a con- 
siderable simplification can be attained in writing down matrix 
elements for complex processes in electrodynamics. Further, a 
physical point of view is available which permits them to be 
written down directly for any specific problem. Being simply a 
restatement of conventional electrodynamics, however, the matrix 
elements diverge for complex processes. (2) Electrodynamics is 
modified by altering the interaction of electrons at short distances. 
All matrix elements are now finite, with the exception of those 
relating to problems of vacuum polarization. The latter are 
evaluated in a manner suggested by Pauli and Bethe, which gives 
finite results for these matrices also. The only effects sensitive to 
the modification are changes in mass and charge of the electrons. 
Such changes could not be directly observed. Phenomena directly 
observable, are insensitive to the details of the modification used 
(except at extreme energies). For such phenomena, a limit can 
be taken as the range of the modification goes to zero. The results 
then agree with those of Schwinger. A complete, unambiguous, 



and presumably consistent, method is therefore available for the 
calculation of all processes involving electrons and photons. 

The simplification in writing the expressions results from an 
emphasis on the over-all space-time view resulting from a study 
of the solution of the equations of electrodynamics. The relation 
of this to the more conventional Hamiltonian point of view is 
discussed. It would be very difficult to make the modification 
which is proposed if one insisted on having the equations in 
Hamiltonian form. 

The methods apply as well to charges obeying the Klein-Gordon 
equation, and to the various meson theories of nuclear forces. 
Illustrative examples are given. Although a modification like that 
used in electrodynamics can make all matrices finite for all of the 
meson theories, for some of the theories it is no longer true that 
all directly observable phenomena are insensitive to the details of 
the modification used. 

The actual evaluation of integrals appearing in the matrix 
elements may be facilitated, in the simpler cases, by methods 
described in the appendix. 



I HIS paper should be considered as a direct con- 
■*■ tinuation of a preceding one 1 (I) in which the 
motion of electrons, neglecting interaction, was ana- 
lyzed, by dealing directly with the solution of the 
Hamiltonian differential equations. Here the same tech- 
nique is applied to include interactions and in that way 
to express in simple terms the solution of problems in 
quantum electrodynamics. 

For most practical calculations in quantum electro- 
dynamics the solution is ordinarily expressed in terms 
of a matrix element. The matrix is worked out as an 
expansion in powers of e 2 /hc, the successive terms cor- 
responding to the inclusion of an increasing number of 
virtual quanta. It appears that a considerable simplifi- 
cation can be achieved in writing down these matrix 
elements for complex processes. Furthermore, each term 
in the expansion can be written down and understood 
directly from a physical point of view, similar to the 
space-time view in I. It is the purpose of this paper to 
describe how this may be done. We shall also discuss 
methods of handling the divergent integrals which 
appear in these matrix elements. 

The simplification in the formulae results mainly from 
the fact that previous methods unnecessarily separated 
into individual terms processes that were closely related 
physically. For example, in the exchange of a quantum 
between two electrons there were two terms depending 
on which electron emitted and which absorbed the 
quantum. Yet, in the virtual states considered, timing 
relations are not significant. Olny the order of operators 
in the matrix must be maintained. We have seen (I), 
that in addition, processes in which virtual pairs are 
produced can be combined with others in which only 

1 R. P. Feynman, Phys. Rev. 76, 749 (1949), hereafter called I. 



positive energy electrons are involved. Further, the 
effects of longitudinal and transverse waves can be 
combined together. The separations previously made 
were on an unrelativistic basis (reflected in the circum- 
stance that apparently momentum but not energy is 
conserved in intermediate states). When the terms are 
combined and simplified, the relativistic invariance of 
the result is self-evident. 

We begin by discussing the solution in space and time 
of the Schrodinger equation for particles interacting 
instantaneously. The results are immediately general- 
izable to delayed interactions of relativistic electrons 
and we represent in that way the laws of quantum 
electrodynamics. We can then see how the matrix ele- 
ment for any process can be written down directly. In 
particular, the self-energy expression is written down. 

So far, nothing has been done other than a restate- 
ment of conventional electrodynamics in other terms. 
Therefore, the self-energy diverges. A modification 2 in 
interaction between charges is next made, and it is 
shown that the self-energy is made convergent and 
corresponds to a correction to the electron mass. After 
the mass correction is made, other real processes are 
finite and insensitive to the "width" of the cut-off in 
the interaction. 3 

Unfortunately, the modification proposed is not com- 
pletely satisfactory theoretically (it leads to some diffi- 
culties of conservation of energy). It does, however, 
seem consistent and satisfactory to define the matrix 



2 For a discussion of this modification in classical physics see 
R. P. Feynman, Phys. Rev. 74 939 (1948), hereafter referred 
to as A. 

3 A brief summary of the methods and results will be found in 
R. P. Feynman, Phys. Rev. 74, 1430 (1948), hereafter referred 
to as B. 



769 



179 



770 



FEYNMAN 



element for all real processes as the limit of that com- 
puted here as the cut-off width goes to zero. A similar 
technique suggested by Pauli and by Bethe can be 
applied to problems of vacuum polarization (resulting 
in a renormalization of charge) but again a strict 
physical basis for the rules of convergence is not known. 

After mass and charge renormalization, the limit of 
zero cut-off width can be taken for all real processes. 
The results are then equivalent to those of Schwinger 4 
who does not make explicit use of the convergence fac- 
tors. The method of Schwinger is to identify the terms 
corresponding to corrections in mass and charge and, 
previous to their evaluation, to remove them from the 
expressions for real processes. This has the advantage 
of showing that the results can be strictly independent 
of particular cut-off methods. On the other hand, many 
of the properties of the integrals are analyzed using 
formal properties of invariant propagation functions. 
But one of the properties is that the integrals are infinite 
and it is not clear to what extent this invalidates the 
demonstrations. A practical advantage of the present 
method is that ambiguities can be more easily resolved; 
simply by direct calculation of the otherwise divergent 
integrals. Nevertheless, it is not at all clear that the 
convergence factors do not upset the physical con- 
sistency of the theory. Although in the limit the two 
methods agree, neither method appears to be thoroughly 
satisfactory theoretically. Nevertheless, it does appear 
that we now have available a complete and definite 
method for the calculation of physical processes to any 
order in quantum electrodynamics. 

Since we can write down the solution to any physical 
problem, we have a complete theory which could stand 
by itself. It will be theoretically incomplete, however, 
in two respects. First, although each term of increasing 
order in e 2 /hc can be written down it would be desirable 
to see some way of expressing things in finite form to 
all orders in e 2 /hc at once. Second, although it will be 
physically evident that the results obtained are equiva- 
lent to those obtained by conventional electrodynamics 
the mathematical proof of this is not included. Both of 
these limitations will be removed in a subsequent paper 
(see also Dyson 4 ). 

Briefly the genesis of this theory was this. The con- 
ventional electrodynamics was expressed in the La- 
grangian form of quantum mechanics described in the 
Reviews of Modern Physics. 5 The motion of the field 
oscillators could be integrated out (as described in Sec- 
tion 13 of that paper), the result being an expression of 
the delayed interaction of the particles. Next the modi- 
fication of the delta-function interaction could be made 
directly from the analogy to the classical case. 2 This 



4 J. Schwinger, Phys. Rev. 74, 1439 (1948), Phys. Rev. 75, 651 
(1949). A proof of this- equivalence is given by F. J. Dyson, Phys. 
Rev. 75, 486 (1949). 

5 R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). The applica- 
tion to electrodynamics is described in detail by H. J. Groenewold, 
Koninklijke Nederlandsche Akademia van Weteschappen. Pro- 
ceedings Vol. LII, 3 (226) 1949. 



was still not complete because the Lagrangian method 
had been worked out in detail only for particles obeying 
the non-relativistic Schrodinger equation. It was then 
modified in accordance with the requirements of the 
Dirac equation and the phenomenon of pair creation. 
This was made easier by the reinterpretation of the 
theory of holes (I). Finally for practical calculations the 
expressions were developed in a power series in e 2 /hc. It 
was apparent that each term in the series had a simple 
physical interpretation. Since the result was easier to 
understand than the derivation, it was thought best to 
publish the results first in this paper. Considerable time' 
has been spent to make these first two papers as com- 
plete and as physically plausible as possible without 
relying on the Lagrangian method, because it is not 
generally familiar. It is realized that such a description 
cannot carry the conviction of truth which would ac- 
company the derivation. On the other hand, in the 
interest of keeping simple things simple the derivation 
will appear in a separate paper. 

The possible application of these methods to the 
various meson theories is discussed briefly. The formu- 
las corresponding to a charge particle of zero spin 
moving in accordance with the Klein Gordon equation 
are also given. In an Appendix a method is given for 
calculating the integrals appearing in the matrix ele- 
ments for the simpler processes. 

The point of view which is taken here of the inter- 
action of charges differs from the more usual point of 
view of field theory. Furthermore, the familiar Hamil- 
tonian form of quantum mechanics must be compared 
to the over-all space-time view used here. The first 
section is, therefore, devoted to a discussion of the 
relations of these viewpoints. 

1. COMPARISON WITH THE HAMILTONIAN 
METHOD 

Electrodynamics can be looked upon in two equiva- 
lent and complementary ways. One is as the description 
of the behavior of a field (Maxwell's equations). The 
other is as a description of a direct interaction at a 
distance (albeit delayed in time) between charges (the 
solutions of Lienard and Wiechert). From the latter 
point of view light is considered as an interaction of the 
charges in the source with those in the absorber. This is 
an impractical point of view because many kinds of 
sources produce the same kind of effects. The field point 
of view separates these aspects into two simpler prob- 
lems, production of light, and absorption of light. On 
the other hand, the field point of view is less practical 
when dealing with close collisions of particles (or their 
action on themselves). For here the source and absorber 
are not readily distinguishable, there is an intimate 
exchange of quanta. The fields are so closely determined 
by the motions of the particles that it is just as well not 
to separate the question into two problems but to con- 
sider the process as a direct interaction. Roughly, the 
field point of view is most practical for problems involv- 



180 



QUANTUM ELECTRODYNAMICS 



771 



ing real quanta, while the interaction view is best for 
the discussion of the virtual quanta involved. We shall 
emphasize the interaction viewpoint in this paper, first 
because it is less familiar and therefore requires more 
discussion, and second because the important aspect in 
the problems with which we shall deal is the effect of 
virtual quanta. 

The Hamiltonian method is not well adapted to 
represent the direct action at a distance between charges 
because that action is delayed. The Hamiltonian method 
represents the future as developing out of the present. 
If the values of a complete set of quantities are known 
now, their values can be computed at the next instant 
in time. If particles interact through a delayed inter- 
action, however, one cannot predict the future by 
simply knowing the present motion of the particles. 
One would also have to know what the motions of the 
particles were in the past in view of the interaction this 
may have on the future motions. This is done in the 
Hamiltonian electrodynamics, of course, by requiring 
that one specify besides the present motion of the 
particles, the values of a host of new variables (the 
coordinates of the field oscillators) to keep track of that 
aspect of the past motions of the particles which de- 
termines their future behavior. The use of the Hamil- 
tonian forces one to choose the field viewpoint rather 
than the interaction viewpoint. 

In many problems, for example, the close collisions 
of particles, we are not interested in the precise tem- 
poral sequence of events. It is not of interest to be able 
to say how the situation would look at each instant of 
time during a collision and how it progresses from in- 
stant to instant. Such ideas are only useful for events 
taking a long time and for which we can readily obtain 
information during the intervening period. For collisions 
it is much easier to treat the process as a whole. 6 The 
M0ller interaction matrix for the the collision of two elec- 
trons is not essentially more complicated than the non- 
relativistic Rutherford formula, yet the mathematical 
machinery used to obtain the former from quantum 
electrodynamics is vastly more complicated than 
Schrodinger's equation with the e 2 /rn interaction 
needed to obtain the latter. The difference is only that 
in the latter the action is instantaneous so that the 
Hamiltonian method requires no extra variables, while 
in the former relativistic case it is delayed and the 
Hamiltonian method is very cumbersome. 

We shall be discussing the solutions of equations 
rather than the time differential equations from which 
they come. We shall discover that the solutions, because 
of the over-all space-time view that they permit, are as 
easy to understand when interactions are delayed as 
when they are instantaneous. 

As a further point, relativistic invariance will be self- 
evident. The Hamiltonian form of the equations de- 
velops the future from the instantaneous present. But 

6 This is the viewpoint of the theory of the 5 matrix of Heisen- 
berg. 



for different observers in relative motion the instan- 
taneous present is different, and corresponds to a 
different 3-dimensional cut of space-time. Thus the 
temporal analyses of different observers is different and 
their Hamiltonian equations are developing the process 
in different ways. These differences are irrelevant, how- 
ever, for the solution is the same in any space time 
frame. By forsaking the Hamiltonian method, the 
wedding of relativity and quantum mechanics can be 
accomplished most naturally. 

We illustrate these points in the next section by 
studying the solution of Schrodinger's equation for non- 
relativistic particles interacting by an instantaneous 
Coulomb potential (Eq. 2). When the solution is modi- 
fied to include the effects of delay in the interaction 
and the relativistic properties of the electrons we obtain 
an expression of the laws of quantum electrodynamics 
(Eq. 4). 

2. THE INTERACTION BETWEEN CHARGES 
We study by the same methods as in I, the interaction 
of two particles using the same notation as I. We start 
by considering the non-relativistic case described by the 
Schrodinger equation (I, Eq. 1). The wave function at 
a given time is a function ip(x a , x b , t) of the coordinates 
x a and x b of each particle. Thus call A'(x a , x b , /; x a ', x b , /') 
the amplitude that particle a at x a ' at time /' will get 
to x a at / while particle b at x b ' at t' gets to x fc at /. If the 
particles are free and do not interact this is 

A^(x a , x 6 , /; x a ', x b , i') = K 0a (x a , t; x a ', t')K 0b (x b , (; x b ', /') 

where K 0a is the A' function for particle a considered 
as free. In this case we can obviously define a quantity 
like K, but for which the time / need not be the same 
for particles a and b (likewise for t'); e.g., 

K (3, 4; 1, 2) = K 0a (3, 1)£ 06 (4, 2) (1) 

can be thought of as the amplitude that particle a goes 
from Xi at t\ to x 3 at / 3 and that particle b goes from x 2 
at t 2 to x 4 at t A . 

When the particles do interact, one can only define 
the quantity K(3, 4; 1, 2) precisely if the interaction 
vanishes between ti and t% and also between tz and U. 
In a real physical system such is not the case. There is 
such an enormous advantage, however, to the concept 
that we shall continue to use it, imagining that we can 
neglect the effect of interactions between t x and h and 
between / 3 and t^ For practical problems this means 
choosing such long time intervals / 3 — h and i 4 — h that 
the extra interactions near the end points have small 
relative effects. As an example, in a scattering problem 
it may well be that the particles are so well separated 
initially and finally that the interaction at these times 
is negligible. Again energy values can be defined by the 
average rate of change o'f phase over such long time 
intervals that errors initially and finally can be neg- 
lected. Inasmuch as any physical problem can be defined 
in terms of scattering processes we do not lose much in 



181 

772 R.P.FEYNMAN 




Fig. 1. The fundamental interaction Eq. (4). Exchange of one 
quantum between two electrons. 

a general theoretical sense by this approximation. If it 
is not made it is not easy to study interacting particles 
relativistically, for there is nothing significant in choos- 
ing t x =tz if Xi^x 3 , as absolute simultaneity of events 
at a distance cannot be defined invariantly. It is essen- 
tially to avoid this approximation that the complicated 
structure of the older quantum electrodynamics has 
been built up. We wish to describe electrodynamics as 
a delayed interaction between particles. If we can make 
the approximation of assuming a meaning to K (3, 4; 1, 2) 
the results of this interaction can be expressed very 
simply. 

To see how this may be done, imagine first that the 
interaction is simply that given by a Coulomb potential 
#/r where r is the distance between the particles. If this 
be turned on only for a very short time A/ at time t , 
the first order correction to if (3, 4; 1, 2) can be worked 
out exactly as was Eq. (9) of I by an obvious general- 
ization to two particles: 

K»(3, 4; 1 ,2) = -ie 2 f f ' K 0a (3, 5)A' 0b (4, 6)r 66 " 1 

XA' 0a (5, l)A' 0h (6, 2)<Z 3 x 5 d 3 x 6 A* , 

where t b = l 6 = t . If now the potential were on at all 
times (so that strictly K is not defined unless ti = t 3 and 
h=h), the first-order effect is obtained by integrating 
on to, which we can write as an integral over both t 6 
and h if we include a delta- function 5(/ 5 — to) to insure 
contribution only when t b = t 6 . Hence, the first-order 
effect of interaction is (calling t b — t 6 =t b6 ): 

K™(3, 4; 1, 2)=-ie 2 f f K Qa (3, 5)K 0b (4, 6)r 66 - ] 

X5(t b ,)K 0a (5, l)K ob (6, 2)dT b dr 6 , (2) 

where dr = d 3 xdt. 

We know, however, in classical electrodynamics, that 
the Coulomb potential does not act instantaneously, 
but is delayed by a time r 56 , taking the speed of light 
as unity. This suggests simply replacing r 6 6 _1 5(/56) in 
(2) by something like r 5 6 _1 5(/ 5 6— r b6 ) to represent the 
delay in the effect of b on a. 



This turns out to be not quite right, 7 for when this 
interaction is represented by photons they must be of 
only positive energy, while the Fourier transform of 
S(t b 6— r b6 ) contains frequencies of both signs. It should 
instead be replaced by 5+(/ 56 — r 56 ) where 



5+0*0= I e-' ux du/TT = \im =5(x)-\-( 

J Q «~o x-ie 



(3) 



This is to be averaged with r 56 -1 5+(— / 5 6— r 5 e) which 
arises when t b <lo and corresponds to a emitting the 
quantum which b receives. Since 

(2r)->(5+0-r) + 5+(-/-r)) = 5 + (/ 2 -r 2 ), 

this means r bf r l 8(t b e) is replaced by 5 + (s 56 2 ) where 
s b o 2 = t bb 2 — r bf ? is the square of the relativistically in- 
variant interval between points 5 and 6. Since in 
classical electrodynamics there is also an interaction 
through the vector potential, the complete interaction 
(see A, Eq. (1)) should be (1 — (v 5 -v 6 )5+(5 5 6 2 ), or in the 
relativistic case, 

(1 — a a - u b )5 + (s b <f) = (3 a j3 b y aiJi y bli 5 + (s b <r). 

Hence we have for electrons obeying the Dirac equation, 



KW(3, 4; 1,2) = - ie 2 



J> 



,(3,5)A' +b (4,6) ToM7 ^ 



XS+(*56 2 )£+a(S, l)A' +b (6, 2)dr s dTo, (4) 

where y ai i and 7 hM are the Dirac matrices applying to 
the spinor corresponding to particles a and b, respec- 
tively (the factor (3 a (3 b being absorbed in the definition, 
I Eq. (17), of K+). 

This is our fundamental equation for electrodynamics. 
It describes the effect of exchange of one quantum 
(therefore first order in e 2 ) between two electrons. It 
will serve as a prototype enabling us to write down the 
corresponding quantities involving the exchange of two 
or more quanta between two electrons or the interaction 
of an electron with itself. It is a consequence of con- 
ventional electrodynamics. Relativistic invariance is 
clear. Since one sums over // it contains the effects of 
both longitudinal and transverse waves in a relati- 
vistically symmetrical way. 

We shall now interpret Eq. (4) in a manner which 
will permit us to write down the higher order terms. It 
can be understood (see Fig. 1) as saying that the ampli- 
tude for "a" to go from 1 to 3 and "6" to go from 2 to 4 
is altered to first order because they can exchange a 
quantum. Thus, "a" can go to 5 (amplitude A^+(5, 1)) 



7 It, and a like term for the effect of a on b, leads to a theory 
which, in the classical limit, exhibits interaction through half- 
advanced and half-retarded potentials. Classically, this is equi- 
valent to purely retarded effects within a closed box from which 
no light escapes (e.g., see A, or J. A. Wheeler and R. P. Feynman, 
Rev. Mod. Phys. 17, 157 (1945)). Analogous theorems exist in 
quantum mechanics but it would lead us too far astray to discuss 
them now. 



182 

QUANTUM ELECTRODYNAMICS 773 



emit a quantum (longitudinal, transverse, or scalar 
y ail ) and then proceed to 3 (K + (3, 5)). Meantime "b" 
goes to 6 (K+(6, 2)), absorbs the quantum (yb») and 
proceeds to 4 (#+(4, 6)). The quantum meanwhile pro- 
ceeds from 5 to 6, which it does with amplitude 5 + (s 66 2 ). 
We must sum over all the possible quantum polariza- 
tions n and positions and times of emission 5, and of 
absorption 6. Actually if t b >t 6 it would be better to 
say that "a" absorbs and "b" emits but no attention 
need be paid to these matters, as all such alternatives 
are automatically contained in (4). 

The correct terms of higher order in e 2 or involving 
larger numbers of electrons (interacting with themselves 
or in pairs) can be written down by the same kind of 
reasoning. They will be illustrated by examples as we 
proceed. In a succeeding paper they will all be deduced 
from conventional quantum electrodynamics. 

Calculation, from (4), of the transition element be- 
tween positive energy free electron states gives the 
Moller scattering of two electrons, when account is 
taken of the Pauli principle. 

The exclusion principle for interacting charges is 
handled in exactly the same way as for non-interacting 
charges (I). For example, for two charges it requires 
only that one calculate K(3, 4; 1, 2) -#(4, 3; 1,2) to 
get the net amplitude for arrival of charges at 3 and 4. 
It is disregarded in intermediate states. The inter- 
ference effects for scattering of electrons by positrons 
discussed by Bhabha will be seen to result directly in 
this formulation. The formulas are interpreted to apply 
to positrons in the manner discussed in I. 

As our primary concern will be for processes in which 
the quanta are virtual we shall not include here the 
detailed analysis of processes involving real quanta in 
initial or final state, and shall content ourselves by only 
stating the rules applying to them. 8 The result of the 
analysis is, as expected, that they can be included by 
the same line of reasoning as is used in discussing the 
virtual processes, provided the quantities are normalized 
in the usual manner to represent single quanta. For 
example, the amplitude that an electron in going from 1 
to 2 absorbs a quantum whose vector potential, suitably 
normalized, is c M exp(— ik-x) = C li (x) is just the expres- 
sion (I, Eq. (13)) for scattering in a potential with 
A (3) replaced by C (3). Each quantum interacts only 



8 Although in the expressions stemming from (4) the quanta are 
virtual, this is not actually a theoretical limitation. One way to 
deduce the correct rules for real quanta from (4) is to note that 
in a closed system all quanta can be considered as virtual (i.e., 
they have a known source and are eventually absorbed) so that 
in such a system the present description is complete and equiva- 
lent to the conventional one. In particular, the relation of the 
Einstein A and B coefficients can be deduced. A more practical 
direct deduction of the expressions for real quanta will be given 
in the subsequent paper. It might be noted that (4) can be re- 
written as describing the action on a, K a) (3, l) = iJ'K + (3, 5) 
XA(5)K+(5, \)dn of the potential 4 M (5) = e 2 /"A' + (4, 6)8 + (s M i )y )1 
XK + (6,2)dTt arising from Maxwell's equations — O i A ll = iirj ll 
from a "current" Jn(6) =e?K + (4, 6)y^K+(6, 2) produced by par- 
ticle b in going from 2 to 4. This is virtue of the fact that 8 + 
satisfies 

- D^+^i 2 ) = 4*5(2, 1). (5) 



once (either in emission or in absorption), terms like 
(I, Eq. (14)) occur only when there is more than one 
quantum involved. The Bose statistics of the quanta 
can, in all cases, be disregarded in intermediate states. 
The only effect of the statistics is to change the weight 
of initial or final states. If there are among quanta, in 
the initial state, some n which are identical then the 
weight of the state is (1/w!) of what it would be if these 
quanta were considered as different (similarly for the 
final state). 

3. THE SELF-ENERGY PROBLEM 

Having a term representing the mutual interaction 
of a pair of charges, we must include similar terms to 
represent the interaction of a charge with itself. For 
under some circumstances what appears to be two dis- 
tinct electrons may, according to I, be viewed also as 
a single electron (namely in case one electron was 
created in a pair with a positron destined to annihilate 
the other electron). Thus to the interaction between 
such electrons must correspond the possibility of the 
action of an electron on itself. 9 

This interaction is the heart of the self energy prob- 
lem. Consider to first order in e 2 the action of an electron 
on itself in an otherwise force free region. The amplitude 
K(2, 1) for a single particle to get from 1 to 2 differs 
from K + {2, 1) to first order in e 2 by a term 

k«(2, 1)= -ie 2 f lK + (2, 4) 7m # + (4, 3J 7 „ 

XK+(3, l)dT- 3 d u 5,(s iZ 2 ). (6) 

It arises because the electron instead of going from 1 
directly to 2, may go (Fig. 2) first to 3, (K + (3, 1)), emit 
a quantum ( 7/( ), proceed to 4, (if+(4, 3)), absorb it 
( 7m ), and finally arrive at 2 (K + (2, 4)). The quantum 
must go from 3 to 4 (5 + (s 4 3 2 )). 

This is related to the self-energy of a free electron in 
the following manner. Suppose initially, time t h we have 
an electron in state /(l) which we imagine to be a posi- 
tive energy solution of Dirac's equation for a free par- 
ticle. After a long time h—t\ the perturbation will alter 



Fig. 2. Interaction of an elec- 
tron with itself, Eq. (6). 




9 These considerations make it appear unlikely that the con- 
tention of J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 
17, 157 (1945), that electrons do not act on themselves, will be a 
successful concept in quantum electrodynamics. 



183 



774 



P. FEYNMAN 



the wave function, which can then be looked upon as 
a superposition of free particle solutions (actually it 
only contains /). The amplitude that g(2) is contained 
is calculated as in (I, Eq. (21)). The diagonal element 
(g=f) is therefore 



//• 



f(2)(3KU(2, 1)^/(1)^x^X2 



(7) 



The time interval T = l 2 —t 1 (and the spatial volume V 
over which one integrates) must be taken very large, 
for the expressions are only approximate (analogous to 
the situation for two interacting charges). 10 This is 
because, for example, we are dealing incorrectly with 
quanta emitted just before t 2 which would normally be 
reabsorbed at times after t 2 . 

If K m (2, 1) from (6) is actually substituted into (7) 
the surface integrals can be performed as was done in 
obtaining I, Eq. (22) resulting in 



*//■ 



/(4) Tm K + (4, 3h li f(3)8 + (s i J)dudu. (8) 



Putting for /(l) the plane wave u exp(— ip-xi) where 
pt is the energy (p 4 ) and momentum of the electron 
(J) 2 = m 2 ), and u is a constant 4-index symbol, (8) 
becomes 



■te 2 



ff(u yii K + (4, 3) 7 „«) 

X exp(i/> • (x 4 — x 3 )) 8 + (s i3 2 )dT 3 dTi, 



the integrals extending over the volume V and time 
interval T. Since if + (4, 3) depends only on the difference 
of the coordinates of 4 and 3, x 43m , the integral on 4 
gives a result (except near the surfaces of the region) 
independent of 3. When integrated on 3, therefore, the 
result is of order VT. The effect is proportional to V, 
for the wave functions have been normalized to unit 



MOMENTUM p-k, 



FACTOR (p-k-m) 



INTERACTION ,lL 




MOMENTUM k, 
FACTOR k~ 2 



MOMENTUM p 



Fig. 3. Interaction of an electron with itself. 
Momentum space, Eq. (11). 



10 This is discussed in reference 5 in which it is pointed out that 
the concept of a wave function loses accuracy if there are delayed 
self-actions. 



volume. If normalized to volume V, the result would 
simply be proportional to T. This is expected, for if the 
effect were equivalent to a change in energy AE, the 
amplitude for arrival in / at t 2 is altered by a factor 
exp( — iAE{t 2 — ti)), or to first order by the difference 
— i(AE)T. Hence, we have 



AE 



= e 2 I («7 M iiC + (4, 3)7^) exp(^-x«)5 + (543 2 )rfT 4 , 



(9) 



integrated over all space-time dr 4 . This expression will 
be simplified presently. In interpreting (9) we have 
tacitly assumed that the wave functions are normalized 
so that (u*u)= (uyiu) = l. The equation may therefore 
be made independent of the normalization by writing 
the left side as (AE)(uyw), or since (uyiu) = (E/m)(uu) 
and mAm = EAE, as Am(uu) where Am is an equivalent 
change in mass of the electron. In this form invariance 
is obvious. 

One can likewise obtain an expression for the energy- 
shift for an electron in a hydrogen atom. Simply replace 
K+ in (8), by K+ iV) , the exact kernel for an electron in 
the potential, V=fie 2 /r, of the atom, and / by a wave 
function (of space and time) for an atomic state. In 
general the AE which results is not real. The imaginary 
part is negative and in exp( — iAET) produces an ex- 
ponentially decreasing amplitude with time. This is 
because we are asking for the amplitude that an atom 
initially with no photon in the field, will still appear 
after time T with no photon. If the atom is in a state 
which can radiate, this amplitude must decay with 
time. The imaginary part of AE when calculated does 
indeed give the correct rate of radiation from atomic 
states. It is zero for the ground state and for a free 
electron. 

In the non-relativistic region the expression for AE 
can be worked out as has been done by Bethe. 11 In the 
relativistic region (points 4 and 3 as close together as a 
Compton wave-length) the K + (V) which should appear 
in (8) can be replaced to first order in V by K + plus 
K+ (l) (2, 1) given in I, Eq. (13). The problem is then 
very similar to the radiationless scattering problem 
discussed below. 

4. EXPRESSION IN MOMENTUM AND 
ENERGY SPACE 

The evaluation of (9), as well as all the other more 
complicated expressions arising in these problems, is 
very much simplified by working in the momentum and 
energy variables, rather than space and time. For this 
we shall need the Fourier Transform of 5 + (s 2 i 2 ) which is 



5 + (W): 



1 I exp(— ik- 



x 21 )k- 2 d% (10) 



which can be obtained from (3) and (5) or from I, 
Eq. (32) noting that 7+(2, 1) for m 2 = is 5+(s 21 2 ) from 



11 H. A. Bethe, Phys. Rev. 72, 339 (1947). 



184 

444W4-W4WWW44444W44444-4W4WW4-44-4-W4-444- 4"9"9-4- 4-9-&J-4. 4.4. 

QUANTUM ELECTRODYNAMICS 775 





a. Eql2 b. Eql3 c. Eq.14 

Fig. 4. Radiative correction to scattering, momentum space. 

I, Eq. (34). The k~ 2 means (k-k)' 1 or more precisely 
the limit as 5— >0 of (k-k+id)' 1 . Further d 4 k means 
(liry-dkidk-idkzdki. If we imagine that quanta are par- 
ticles of zero mass, then we can make the general rule 
that all poles are to be resolved by considering the 
masses of the particles and quanta to have infinitesimal 
negative imaginary parts. 

Using these results we see that the self-energy (9) is 
the matrix element between u and u of the matrix 



{e 2 /iri) 



jy,(p- 



■k-m)- i y l JrH% 



en: 



where we have used the expression (I, Eq. (31)) for the 
Fourier transform of K + . This form for the self-energy 
is easier to work with than is (9). 

The equation can be understood by imagining (Fig. 3) 
that the electron of momentum p emits (7^) a quantum 
of momentum k, and makes its way now with mo- 
mentum p— k to the next event (factor {p—k—m)" 1 ) 
which is to absorb the quantum (another 7 M ). The 
amplitude of propagation of quanta is k~ 2 . (There is a 
factor e 2 /iri for each virtual quantum). One integrates 
over all quanta. The reason an electron of momentum p 
propagates as \/(p—m) is that this operator is the re- 
ciprocal of the Dirac equation operator, and we are 
simply solving this equation. Likewise light goes as 
l/k 2 , for this is the reciprocal D'Alembertian operator 
of the wave equation of light. The first y„ represents 
the current which generates the vector potential, while 
the second is the velocity operator by which this poten- 
tial is multiplied in the Dirac equation when an external 
field acts on an electron. 

Using the same line of reasoning, other problems may 
be set up directly in momentum space. For example, 
consider the scattering in a potential A = A ll y^ varying 
in space and time as a exp(-iq-x). An electron initially 
in state of momentum pi = pi^.y li will be deflected to 
state pi where Pi = pi-\-q. The zero-order answer is 
simply the matrix element of a between states 1 and 2. 
We next ask for the first order (in e 2 ) radiative correc- 
tion due to virtual radiation of one quantum. There are 
several ways this can happen. First for the case illus- 




(a) (b) 

Fig. 5. Compton scattering, Eq. (15). 
trated in Fig. 4(a), find the matrix: 

{e 2 /m) fy^po-k-my'a^-k-my^y^k-Wk. (12) 

For in this case, first 12 a quantum of momentum k is 
emitted (7 M ), the electron then having momentum 
pi— k and hence propagating with factor {pi—k—m)~ l . 
Next it is scattered by the potential (matrix a) receiving 
additional momentum q, propagating on then (factor 
(p2—k—m)~ l ) with the new momentum until the quan- 
tum is reabsorbed (7^). The quantum propagates from 
emission to absorption (k~ 2 ) and we integrate over all 
quanta (d 4 k), and sum on polarization fi. When this is 
integrated on & 4 , the result can be shown to be exactly 
equal to the expressions (16) and (17) given in B for 
the same process, the various terms coming from resi- 
dues of the poles of the integrand (12). 

Or again if the quantum is both emitted and re- 
absorbed before the scattering takes place one finds 
(Fig. 4(b)) 



»/ 



(e 2 /iri) a(/> 1 -.ra)- 1 7„(£i-fe-w)- 1 7„fe- 2 (2 4 Jk, (13) 



or if both emission - and absorption occur after the 
scattering, (Fig. 4(c)) 



(e 2 A0 f 7„(/> 2 - k-my^ipi- 



~ l ak~ 2 d 4 k. (14) 



These terms are discussed in detail below. 

We have now achieved our simplification of the form 
of writing matrix elements arising from virtual proc- 
esses. Processes in which a number of real quanta is 
given initially and finally offer no problem (assuming 
correct normalization). For example, consider the 
Compton effect (Fig. 5(a)) in which an electron in state 
pi absorbs a quantum of momentum q h polarization 
vector ei M so that its interaction is £ , i M 7 M = ei, and emits 
a second quantum of momentum —q», polarization e 2 
to arrive in final state of momentum pi. The matrix for 

12 First, next, etc., here refer not to the order in true time but to 
the succession of events along the trajectory of the electron. That 
is, more precisely, to the order of appearance of the matrices in 
the expressions. 



185 



776 



FEYNMAN 



this process is e 2 (pi+qi— m)~ l e\. The total matrix for 
the Compton effect is, then, 



e2(pi+qi-my i e 1 +e 1 (J) 1 +q2-m)- 1 e2, 



(15) 



the second term arising because the emission of e 2 may 
also precede the absorption of e x (Fig. 5(b)). One takes 
matrix elements of this between initial and final electron 
states (pi+qi = p2— 92), to obtain the Klein Nishina 
formula. Pair annihilation with emission of two quanta, 
etc., are given by the same matrix, positron states being 
those with negative time component of p. Whether 
quanta are absorbed or emitted depends on whether the 
time component of q is positive or negative. 

5. THE CONVERGENCE OF PROCESSES WITH 
VIRTUAL QUANTA 

These expressions are, as has been indicated, no more 
than a re-expression of conventional quantum electro- 
dynamics. As a consequence, many of them are mean- 
ingless. For example, the self-energy expression (9) or 
(11) gives an infinite result when evaluated. The infinity 
arises, apparently, from the coincidence of the 5-function 
singularities in K + (A, 3) and 8 + (sw 2 )- Only at this point 
is it necessary to make a real departure from conven- 
tional electrodynamics, a departure other than simply 
rewriting expressions in a simpler form. 

We desire to make a modification of quantum electro- 
dynamics analogous to the modification of classical 
electrodynamics described in a previous article, A. 
There the 8(si 2 2 ) appearing in the action of interaction 
was replaced by f(si 2 2 ) where f(x) is a function of small 
width and great height. 

The obvious corresponding modification in the quan- 
tum theory is to replace the 8 + (s 2 ) appearing the 
quantum mechanical interaction by a new function 
f+(s 2 ). We can postulate that if the Fourier trans- 
form of the classical /(S12 2 ) is the integral over all k of 
F{k 2 ) exp(—ik -Xi2)d 4 k, then the Fourier transform of 
f + (s 2 ) is the same integral taken over only positive fre- 
quencies ki for t 2 >ti and over only negative ones for 
h<h in analogy to the relation of 8+(s 2 ) to 8(s 2 ). The 
function f(s 2 ) = f(x-x) can be written* as 

f(x-x) = (2ir)- 2 C f sin(£ 4 |* 4 |) 

Xcos(K-x)dk i d 3 Kg(k- k), 

where g{k ■ k) is kc 1 times the density of oscillators and 
may be expressed for positive h as (A, Eq. (16)) 



i(k 2 )= f 



(8(k 2 )-8(k 2 -\ 2 ))G(\)d\ 



where Jl x G(X)d\=l and G involves values of X large 
compared to m. This simply means that the amplitude 

* This relation is given incorrectly in A, equation just pre- 
ceding 16. 



for propagation of quanta of momentum k is 

- F + (k 2 ) = 7T- 1 r (ft- 2 - (k 2 - \ 2 )- i )G(\)d\, 

Jo 
rather than ft" 2 . That is, writing F+{k 2 ) = -ir- l kr 2 C(k 2 ), 

-f+(s 12 2 ) = T- 1 f exp(-^-x 12 )/j- 2 C'(ft 2 )rf 4 £. (16) 

Every integral over an intermediate quantum which 
previously involved a factor d 4 k/k 2 is now supplied with 
a convergence factor C{k 2 ) where 



C(ft 2 ) = 



t: 



■\ 2 (k 2 -X 2 )- l G(X)d\. 



(17) 



The poles are defined by replacing k 2 by k 2 +i8 in the 
limit 5— >0. That is X 2 may be assumed to have an infini- 
tesimal negative imaginary part. 

The function f+(sn 2 ) may still have a discontinuity 
in value on the light cone. This is of no influence for the 
Dirac electron. For a particle satisfying the Klein 
Gordon equation, however, the interaction involves 
gradients of the potential which reinstates the 8 func- 
tion if / has discontinuities. The condition that / is to 
have no discontinuity in value on the light cone implies 
k 2 C(k 2 ) approaches zero as k 2 approaches infinity. In 
terms of G(X) the condition is 



J X 2 G(\)d\ = 0. 



(18) 



This condition will also be used in discussing the con- 
vergence of vacuum polarization integrals. 

The expression for the self-energy matrix is now 



{#!■ 



■i) fy»(p-k-m)- l y»k- 2 d*kC(k 2 ), (19) 



which, since C(k 2 ) falls off at least as rapidly as 1/k 2 , 
converges. For practical purposes we shall suppose 
hereafter that C(k 2 ) is simply — \ 2 /(k 2 — X 2 ) implying 
that some average (with weight G(\)d\) over values of 
X may be taken afterwards. Since in all processes the 
quantum momentum will be contained in at least one 
extra factor of the form (p—k—m) -1 representing 
propagation of an electron while that quantum is in 
the field, we can expect all such integrals with their 
convergence factors to converge and that the result of 
all such processes will now be finite and definite (ex- 
cepting the processes with closed loops, discussed below, 
in which the diverging integrals are over the momenta 
of the electrons rather than the quanta). 

The integral of ( 1 9) with C ( ft 2 ) = - X 2 ( k 2 - X 2 ) - 1 noting 
that p 2 — m 2 , X^>m and dropping terms of order m/X, 
is (see Appendix A) 

(e 2 /27r)[4m(ln(X/m)+|)-i!>(ln(X/m) + 5/4)]. (20) 



186 



QUANTUM ELECTRODYNAMICS 



777 



When applied to a state of an electron of momentum p 
satisfying pii = mu, it gives for the change in mass (as 
in B, Eq. (9)) 

Aw = m0 2 /27r)(31n(\/w)+!). (21) 

6. RADIATIVE CORRECTIONS TO SCATTERING 

We can now complete the discussion of the radiative 
corrections to scattering. In the integrals we include the 
convergence factor C(k 2 ), so that they converge for 
large k. Integral (12 I is also not convergent because of 
the well-known infra-red catastrophy. For this reason 
we calculate (as discussed in B) the value of the integral 
assuming the photons to have a small mass X m i n <^;ra<^X. 
The integral (12) becomes 



(«y« 



i)j:7*(P*- 



k—m)~ l a{pi— k—m)' 



Xy,(k 2 - X min 2 )- 1 ^C(fe 2 - X min 2 ), 



which when integrated (see Appendix B) gives (e 2 /2r) 
times 

I 2( In— 1 V 1 ) +6 tanfl 

L V X min /\ tan20/ 

4 r° -I 

H I a tanaJa a 

tan20 J J 

1 29 

+ — (qa-aq) +ra, (22) 

4m sin20 

where {q 2 )* = 2m sin# and we have assumed the matrix to 
operate between states of momentum pi and pi = p\-\-q 
and have neglected terms of order X m i n /m, ra/X, and 
<7 2 /X 2 . Here the only dependence on the convergence 
factor is in the term ra, where 

r = ln(A/w)+9/4-21n(m/X min ). (23) 

As we shall see in a moment, the other terms (13), 
(14) give contributions which just cancel the ra term. 
The remaining terms give for small q, 



/ 1 4g- / m 3\\ 

■ 2 /47r)( —(qa-aq)+—a( In ) ) 

\2m 3m 2 \ X min 8// 



(24) 



which shows the change in magnetic moment and the 
Lamb shift as interpreted in more detail in B. 13 



13 That the result given in B in Eq. (19) was in error was re- 
peatedly pointed out to the author, in private communication, 
by V. F. Weisskopf and J. B. French, as their calculation, com- 
pleted simultaneously with the author's early in 1948, gave a 
different result. French has finally shown that although the ex- 
pression for the radiationless scattering B, Eq. (18) or (24) above 
is correct, it was incorrectly joined onto Bethe's non-relativistic 
result. He shows that the relation ln2& ma x— l = lnX m i n used by the 
author should have been ln2£ ma x— 5/6=lnX m i n . This results in 
adding a term —(1/6) to the logarithm in B, Eq. (19) so that the 
result now agrees with that of J. B. French and V. F. Weisskopf, 



We must now study the remaining terms (13) and 
(14). The integral on k in (13) can be performed (after 
multiplication by C{k 2 )) since it involves nothing but 
the integral (19) for the self-energy and the result is- 
allowed to operate on the initial state tt\, (so that 
piU\ = muy). Hence the factor following a{pi — m)~ l wilt 
be just Am. But, if one now tries to expand \/{p\— w) 
= (pi+m)/(pi 2 —m 2 ) one obtains an infinite result, 
since pi 2 = m 2 . This is, however, just what is expected 
physically. For the quantum can be emitted and ab- 
sorbed at any time previous to the scattering. Such a 
process has the effect of a change in mass of the electron 
in the state 1. It therefore changes the energy by AE 
and the amplitude to first order in AE by —iAE-t where 
t is the time it is acting, which is infinite. That is, the 
major effect of this term would be canceled by the effect 
of change of mass Am. 

The situation can be analyzed in the following 
manner. We suppose that the electron approaching the 
scattering potential a has not been free for an infinite 
time, but at some time far past suffered a scattering by 
a potential b. If we limit our discussion to the effects 
of Ara and of the virtual radiation of one quantum be- 
tween two such scatterings each of the effects will be 
finite, though large, and their difference is determinate. 
The propagation from b to a is represented by a matrix 



a{j)'—m)~ l b, 



(25) 



in which one is to integrate possibly over p' (depending 
on details of the situation). (If the time is long between 
b and a, the energy is very nearly determined so that 
p' 2 is very nearly m 2 .) 

We shall compare the effect on the matrix (25) of the 
virtual quanta and of the change of mass Am. The effect 
of a virtual quantum is 

{er/m) I a(p' — w) _1 7 M (/>'— k — m)~ x 

Xjnip'-my'bk-WkCik 2 ), (26) 

while that of a change of mass can be written 

a(p'-m)- l Am(p'-m)- l b, (27) 

and we are interested in the difference (26)-(27). A 
simple and direct method of making this comparison is 
just to evaluate the integral on k in (26) and subtract 
from the result the expression (27) where Am is given 
in (21). The remainder can be expressed as a multiple 
— r(p' 2 ) of the unperturbed amplitude (25); 

-r(p'*)a(p'-m)- l b. (28) 

This has the same result (to this order) as replacing 
the potentials a and b in (25) by (1 — \r(p" 2 ))a and 



Phys. Rev. 75, 1240 (1949) and N. H. Kroll and W. E. Lamb, 
Phys. Rev. 75, 388 (1949). The author feels unhappily responsible 
for the very considerable delay in the publication of French's 
result occasioned by this error. This footnote is appropriately 
numbered. 



187 



778 



P. FE YN M AN 



{\-\r{p' 2 ))b. In the limit, then, as p' 2 -+m 2 the net 
effect on the scattering is — \ra where r, the limit of 
r(J>' 2 ) as p' 2 —>m 2 (assuming the integrals have an infra- 
red cut-off), turns out to be just equal to that given in 
(23). An equal term — \ra arises from virtual transitions 
after the scattering (14) so that the entire ra term in 
(22) is canceled. 

The reason that r is just the value of (12) when q 2 = 
can also be seen without a direct calculation as follows: 
Let us call p the vector of length m in the direction of 
p' so that if />' 2 = w(l+e) 2 we have p'= (l+e)p and we 
take e as very small, being of order T~ l where T is the 
time between the scatterings b and a. Since (/>'— ra) -1 
= (p'+m)/(p' 2 -m 2 )~(p+m)/2m 2 e, the quantity (25) 
is of order € _1 or T. We shall compute corrections to it 
only to its own order (e -1 ) in the limit e— >0. The term 
(27) can be written approximately 14 as 

(e 2 /iri) I a(p'— w) _1 7 M (/>— k— w) _1 

Xy,{P'-m)- l bk- 2 d'kC{k 2 ), 

using the expression (19) for Am. The net of the two 
effects is therefore approximately 15 

- (e 2 /iri) laip'-m)-^^- k-m)~ l ep(p- k-m)~ l 

X y.ip'-m^bk-WkCik 2 ), 

a term now of order 1/e (since {p'—m)~ l ^{p-\-m) 
\(2ra 2 e) _1 ) and therefore the one desired in the limit. 
Comparison to (28) gives for r the expression 

(J>i+m/2m) I y^pi-k-m^ipinr^ipi—k—m)-* 

X7„fe- 2 d 4 £C(& 2 ). (29) 

The integral can be immediately evaluated, since it 
is the same as the integral (12), but with ^ = 0, for a 
replaced by Pi/m. The result is therefore r-(pi/m) 
which when acting on the state U\ is just r, as p\U\ = mu\. 
For the same reason the term {pi-\-m)/2m in (29) is 
effectively 1 and we are left with — r of (23). 16 

In more complex problems starting with a free elec- 

14 The expression is not exact because the substitution of Aw 
by the integral in (19) is valid only if p operates on a state such 
that p can be replaced by m. The error, however, is of order 
a{p'-m)~ 1 (p—m)(J)'-m)~ 1 b which is a((l+e)p+m) (p-m) 
X((l+t)p+m)p(2e+e i )- i m- i . But since ^=«',we ha.vep(p-m) 
= —m{p—m) = (p—m)p so the net result is approximately 
a(p—m)b/Am i and is not of order 1/e but smaller, so that its effect 
drops out in the limit. 

15 We have used, to first order, the general expansion (valid for 
any operators A, B) 

(A+B^^A-i-A^BA-i+A-tBA-iBA-i 

with A=p— k— m and B=p'—p = tp to expand the difference of 
(/>'- k-m)- 1 and (p- k-m)~ l . 

16 The renormalization terms appearing B, Eqs. (14), (15) when 
translated directly into the present notation do not give twice 
(29) but give this expression with the central pim~ l factor replaced 
by my t /Ei where E\ = p\ v . for /x = 4. When integrated it therefore 
gives ra((pi+m)/2m)(my t /Ei) or ra~ra(myi/Ei)(pi-m)/2m. 
(Since piyt+y t Pi = 2Ei) which gives just ra, since p\Ui = mui. 



tron the same type of term arises from the effects of a 
virtual emission and absorption both previous to the 
other processes. They, therefore, simply lead to the 
same factor r so that the expression (23) may be used 
directly and these renormalization integrals need not 
be computed afresh for each problem. 

In this problem of the radiative corrections to scatter- 
ing the net result is insensitive to the cut-off. This 
means, of course, that by a simple rearrangement of 
terms previous to the integration we could have avoided 
the use of the convergence factors completely (see for 
example Lewis 17 ). The problem was solved in the 
manner here in order to illustrate how the use of such 
convergence factors, even when they are actually un- 
necessary, may facilitate analysis somewhat by remov- 
ing the effort and ambiguities that may be involved in 
trying to rearrange the otherwise divergent terms. 

The replacement of 5+ by /+ given in (16), (17) is 
not determined by the analogy with the classical prob- 
lem. In the classical limit only the real part of 8+ (i.e., 
just 5) is easy to interpret. But by what should the 
imaginary part, l/(7ris 2 ), of 5 + be replaced? The choice 
we have made here (in defining, as we have, the location 
of the poles of (17)) is arbitrary and almost certainly 
incorrect. If the radiation resistance is calculated for 
an atom, as the imaginary part of (8), the result de- 
pends slightly on the function /+. On the other hand the 
light radiated at very large distances from a source is 
independent of /+. The total energy absorbed by distant 
absorbers will not check with the energy loss of the 
source. We are in a situation analogous to that in the 
classical theory if the entire / function is made to 
contain only retarded contributions (see A, Appendix). 
One desires instead the analogue of (F) Tet of A. This 
problem is being studied. 

One can say therefore, that this attempt to find a 
consistent modification of quantum electrodynamics is 
incomplete (see also the question of closed loops, below). 
For it could turn out that any correct form of /+ which 
will guarantee energy conservation may at the same 
time not be able to make the self-energy integral finite. 
The desire to make the methods of simplifying the 
calculation of quantum electrodynamic processes more 
widely available has prompted this publication before 
an analysis of the correct form for /+ is complete. One 
might try to take the position that, since the energy 
discrepancies discussed vanish in the limit X— >°o, the 
correct physics might be considered to be that obtained 
by letting X— >co after mass renormalization. I have no 
proof of the mathematical consistency of this procedure, 
but the presumption is very strong that it is satisfac- 
tory. (It is also strong that a satisfactory form for / + 
can be found.) 

7. THE PROBLEM OF VACUUM POLARIZATION 

In the analysis of the radiative corrections to scatter- 
ing one type of term was not considered. The potential 

17 H. W. Lewis, Phys. Rev. 73, 173 (1948). 



188 



QUANTUM ELECTRODYNAMICS 



779 



which we can assume to vary as a M exp(— iq-x) creates 
a pair of electrons (see Fig. 6), momenta p a , —pb. This 
pair then reannihilates, emitting a quantum q = pb—p a , 
which quantum scatters the original electron from state 
1 to state 2. The matrix element for this process (and 
the others which can be obtained by rearranging the 
order in time of the various events) is 

- («*/«) (fl*y„«i) fsp[(p a +q-m)-i 

Xy.(pa~ m^yJd'Paqr'Ciq 2 )^. (30) 

This is because the potential produces the pair with 
amplitude proportional to a v y v , the electrons of mo- 
menta p a and — (p a -\- q) proceed from there to annihi- 
late, producing a quantum (factor y a ) which propagates 
(factor q~ 2 C(q 2 )) over to the other electron, by which 
it is absorbed (matrix element of y^ between states 1 
and 2 of the original electron (w 2 7 M «i)). All momenta p a 
and spin states of the virtual electron are admitted, 
which means the spur and the integral on d*p a are 
calculated. 

One can imagine that the closed loop path of the 
positron-electron produces a current 



^Jn = Jp 



(31) 



which is the source of the quanta which act on the 
second electron. The quantity 

J,„=-(<?/iri) f Sp[(P+q-myi 

Xy,{p-m)- l y^y,% (32) 

is then characteristic for this problem of polarization 
of the vacuum. 

One sees at once that J ^ diverges badly. The modifi- 
cation of 5 to / alters the amplitude with which the 
current jn will affect the scattered electron, but it can 
do nothing to prevent the divergence of the integral (32) 
and of its effects. 

One way to avoid such difficulties is apparent. From 
one point of view we are considering all routes by which 
a given electron can get from one region of space-time 
to another, i.e., from the source of electrons to the 
apparatus which measures them. From this point of 
view the closed loop path leading to (32) is unnatural. 
It might be assumed that the only paths of meaning are 
those which start from the source and work their way 
in a continuous path (possibly containing many time 
reversals) to the detector. Closed loops would be ex- 
cluded. We have already found that this may be done 
for electrons moving in a fixed potential. 

Such a suggestion must meet several questions, how- 
ever. The closed loops are a consequence of the usual 
hole theory in electrodynamics. Among other things, 
they are required to keep probability conserved. The 
probability that no pair is produced by a potential is 




Fig. 6. Vacuum polarization ef- 
fect on scattering, Eq. (30). p + a 



not unity and its deviation from unity arises from the 
imaginary part of 7 M „. Again, with closed loops ex- 
cluded, a pair of electrons once created cannot annihi- 
late one another again, the scattering of light by light 
would be zero, etc. Although we are not experimentally 
sure of these phenomena, this does seem to indicate 
that the closed loops are necessary. To be sure, it is 
always possible that these matters of probability con- 
servation, etc., will work themselves out as simply in 
the case of interacting particles as for those in a fixed 
potential. Lacking such a demonstration the presump- 
tion is that the difficulties of vacuum polarization are 
not so easily circumvented. 18 

An alternative procedure discussed in B is to assume 
that the function K+(2, 1) used above is incorrect and 
is to be replaced by a modified function K + ' having no 
singularity on the light cone. The effect of this is to 
provide a convergence factor C(p 2 —m 2 ) for every inte- 
gral over electron momenta. 19 This will multiply the 
integrand of (32) by C{f- m 2 )C{{p+q) 2 -m 2 ), since the 
integral was originally 8(p a — pb+ q)d 4 p a d 4 p b and both 
p a and pb get convergence factors. The integral now 
converges but the result is unsatisfactory. 20 

One expects the current (31) to be conserved, that is 
q»j»=0 or q li J liV =0. Also one expects no current if a„ 
is a gradient, or a v = q v times a constant. This leads to 
the condition / M „g„ = which is equivalent to g M / M „ = 
since J M » is symmetrical. But when the expression (32) 
is integrated with such convergence factors it does not 
satisfy this condition. By altering the kernel from K to 
another, K', which does not satisfy the Dirac equation 
we have lost the gauge invariance, its consequent cur- 
rent conservation and the general consistency of the 
theory. 

One can see this best by calculating J ^q, directly 
from (32). The expression within the spur becomes 
(P J rq—>n)~ 1 q(p—m)- 1 y ll which can be written as the 
difference of two terms: (/»— w) _1 7 M — (p+q— ?») -1 Y„- 
Each of these terms would give the same result if the 
integration d*p were without a convergence factor, for 



18 It would be very interesting to calculate the Lamb shift 
accurately enough to be sure that the 20 megacycles expected 
from vacuum polarization are actually present. 

19 This technique also makes self-energy and radiationless scat- 
tering integrals finite even without the modification of 5+ to/ + for 
the radiation (and the consequent convergence factor C(k : ) for 
the quanta). See B. 

20 Added to the terms given below (33) there is a term 
l(\ 3 -2 / j.- + lq' ! )8 tu , for C{k 2 ) = -X^tf-X 2 )" 1 , which is not gauge 
invariant. (In addition the charge renormalization has — 7/6 added 
to the logarithm.) 



189 



780 



R. P. FEYNMAN 



the first can be converted into the second by a shift of 
the origin of p, namely p' = p+q- This does not result 
in cancelation in (32) however, for the convergence 
factor is altered by the substitution. 

A method of making (32) convergent without spoiling 
the gauge invariance has been found by Bethe and by 
Pauli. The convergence factor for light can be looked 
upon as the result of superposition of the effects of 
quanta of various masses (some contributing nega- 
tively). Likewise if we take the factor C(p 2 —m 2 ) 
= -X i (p°—m 2 -X 2 )- i so that (p 2 -m 2 )- l C(p 2 -m 2 ) 
= (p 2 —m 2 )~ l —(p 2 —m 2 —X 2 )~ l we are taking the differ- 
ence of the result for electrons of mass m and mass 
(X 2 +w 2 )*. But we have taken this difference for each 
propagation between interactions with photons. They 
suggest instead that once created with a certain mass 
the electron should continue to propagate with this 
mass through all the potential interactions until it 
closes its loop. That is if the quantity (32), integrated 
over some finite range of p, is called J ^(m 2 ) and the 
corresponding quantity over the same range of p, but 
with m replaced by (ra 2 +X 2 )* is J iit {m 2 -\-X 1 ) we should 
calculate 






)-J li ,(m 2 +X 2 )^G(X)dX, (32') 



the function G(X) satisfying Jl x G(X)dX=l and 
Jl x G(X)X 2 dX = 0. Then in the expression for J M „ P the 
range of p integration can be extended to infinity as the 
integral now converges. The result of the integration 
using this method is the integral on dX over G(X) of 
(see Appendix C) 

X 2 
Av p = 



e 2 / 1 X 2 

— (qtf,-8^q 2 )[ In— 

t \ 3 m- 



[4m 2 +2q 



L 3q 



2 V tan0/ 9JV 



(33) 



with q 2 = 4m 2 sin 2 0. 

The gauge invariance is clear, since q^q^q^—q^S^) = 0. 
Operating (as it always will) on a potential of zero 
divergence the (q»q v — b^q^a, is simply — q 2 a„ the 
D'Alembertian of the potential, that is, the current pro- 
ducing the potential. The term — f (ln(X 2 /ra 2 ))(^„ 

— q 2 8^) therefore gives a current proportional to the 
current producing the potential. This would have the 
same effect as a change in charge, so that we would have 
a difference A(e 2 ) between e 2 and the experimen- 
tally observed charge, e 2 -\-A(e 2 ), analogous to the dif- 
ference between m and the observed mass. This charge 
depends logarithmically on the cut-off, A(e 2 )/e 2 = 

— (2e 2 /3ir) h\(X/m). After this renormalization of charge 
is made, no effects will be sensitive to the cut-off. 

After this is done the final term remaining in (33), 
contains the usual effects 21 of polarization of the vacuum. 

21 E. A. Uehling, Phys. Rev. 48, 55 (1935), R. Serber, Phys. 
Rev. 48, 49 (1935). 



It is zero for a free light quantum (q 2 ~0). For small q z 
it behaves as (2/\S)q l (adding — \ to the logarithm in 
the Lamb effect). For q 2 >(2m) 2 it is complex, the 
imaginary part representing the loss in amplitude re- 
quired by the fact that the probability that no quanta 
are produced by a potential able to produce pairs 
((q 2 ) i >2m) decreases with time. (To make the neces- 
sary analytic continuation, imagine m to have a small 
negative imaginary part, so that (1 — q 2 /4 : m 2 ) i becomes 

— i(q 2 /4:m 2 — l) s as q 2 goes from below to above 4m 2 . 
Then d=ir/2-\-iu where sinhw = + (<7 2 /4w 2 — 1)*, and 

- l/tan0 = i tanhtt = + i (q 2 - Am 2 ) K<7 2 ) -5 -) 

Closed loops containing a number of quanta or poten- 
tial interactions larger than two produce no trouble. 
Any loop with an odd number of interactions gives zero 
(I, reference 9). Four or more potential interactions give 
integrals which are convergent even without a con- 
vergence factor as is well known. The situation is 
analogous to that for self-energy. Once the simple 
problem of a single closed loop is solved there are 
no further divergence difficulties for more complex 
processes. 22 

8. LONGITUDINAL WAVES 

In the usual form of quantum electrodynamics the 
longitudinal and transverse waves are given separate 
treatment. Alternately the condition (dAJdx l ,) < if = Q is 
carried along as a supplementary condition. In the 
present form no such special considerations are neces- 
sary for we are dealing with the solutions of the equation 
— I \ 2 A u =4:irj u with a current _/ M which is conserved 
dj fl /dx^ = 0. That means at least □ ! (6M ;1 /dx M ) = and 
in fact our solution also satisfies 6M M /dx M =0. 

To show that this is the case we consider the ampli- 
tude for emission (real or virtual) of a photon and show 
that the divergence of this amplitude vanishes. The 
amplitude for emission for photons polarized in the ^ 
direction involves matrix elements of y m . Therefore 
what we have to show is that the corresponding matrix 
elements of q^y^q vanish. For example, for a first 
order effect we would require the matrix element of q 
between two states pi and p2=pr\-q. But since 
q = p2~pi and (u2piUi) = m(u2iii) = (u2p2Ui) the matrix 
element vanishes, which proves the contention in this 
case. It also vanishes in more complex situations (essen- 
tially because of relation (34), below) (for example, try 
putting e 2 = <72 in the matrix (15) for the Compton 
Effect). 

To prove this in general, suppose d, i= 1 to N are a 
set of plane wave disturbing potentials carrying mo- 
menta qi (e.g., some may be emissions or absorptions of 
the same or different quanta) and consider a matrix for 
the transition from a state of momentum p to px such 

22 There are loops completely without external interactions. For 
example, a pair is created virtually along with a photon. Next they 
annihilate, absorbing this photon. Such loops are disregarded on 
the grounds that they do not interact .with anything and are 
thereby completely unobservable. Any indirect effects they may 
have via the exclusion principle have already been included. 



190 



QUANTUM ELECTRODYNAMICS 



781 



as as ITt=i Ar_1 (Pi— «*) _1 a,- where />,=/> i_i+ Qi (and in the 
product, terms with larger i are written to the left). 
The most general matrix element is simply a linear 
combination of these. Next consider the matrix be- 
tween states p and px+Q in a situation in which not 
only are the a* acting but also another potential 
a exp( — iq • x) where a = q. This may act previous to all a,, 
in which case it gives CLNWipi+q— m)~ l ai(p +q— m)~ l q 
which is equivalent to +aNYl(pi+q—m)~ l ai since 
-\-{po-\-q — m)~ l q is equivalent to (po+q— m)~ l 
Xipo+q—m) as p is equivalent to m acting on the 
initial state. Likewise if it acts after all the potentials 
it gives q{pN~ fn)~ l a N J\(pi— w) -l fl» which is equivalent 
to — ajvII(/ > >~ m)~ l ai since pN-\-q—m gives zero on the 
final state. Or again it may act between the potential 
a k and a k +\ for each k. This gives 

£flif II {pi+q-m)- l a t {p k +q-m)- 1 

Ar=l i=Ap+l 



Xqipk-my^kYL (pj-m)- l aj. 



However, 



(Pk+ Q - m)- y q{p k - m)~ l 

= (p k -m)- l -(p k +q-m)-\ (34) 

so that the sum breaks into the difference of two sums, 
the first of which may be converted to the other by the 
replacement of k by k—l. There remain only the terms 
from the ends of the range of summation, 

JV-l N-l 

+ a N II {pi-m)- x ai-a N \\ (pi+q-m)- 1 ^. 

t=l i=l 

These cancel the two terms originally discussed so that 
the entire effect is zero. Hence any wave emitted will 
satisfy dAJdx^—0. Likewise longitudinal waves (that 
is, waves for which A^dcp/dx^ or a=q) cannot be 
absorbed and will have no effect, for the matrix ele- 
ments for emission and absorption are similar. (We 
have said little more than that a potential A il =d<p/dx lt 
has no effect on a Dirac electron since a transformation 
^' = exp( — i<p)\p removes it. It is also easy to see in 
coordinate representation using integrations by parts.) 
This has a useful practical consequence in that in 
computing probabilities for transition for unpolarized 
light one can sum the squared matrix over all four 
directions rather than just the two special polarization 
vectors. Thus suppose the matrix element for some 
process for light polarized in direction e M is e^M^. If the 
light has wave vector q^ we know from the argument 
above that q^M ll = 0. For unpolarized light progress- 
ing in the z direction we would ordinarily calculate 
M x 2 -\-M y 2 . But we can as well sum M x 2 -\- M y 2J r M 2 2 — M t 2 
for q^Mp implies M t = M z since qt = q z for free quanta. 
This shows that unpolarized light is a relativistically 
invariant concept, and permits some simplification in 
computing cross sections for such light. 



Incidentally, the virtual quanta interact through 
terms like y m - • -y^hrH^k. Real processes correspond to 
poles in the formulae for virtual processes. The pole 
occurs when k 2 — 0, but it looks at first as though in the 
sum on all four values of n, of y m • • • y^ we would have 
four kinds of polarization instead of two. Now it is clear 
that only two perpendicular to k are effective. 

The usual elimination of longitudinal and scalar vir- 
tual photons (leading to an instantaneous Coulomb 
potential) can of course be performed here too (although 
it is not particularly useful). A typical term in a virtual 
transition is y ,,- • ■ y y.krH^k where the ••• represent 
some intervening matrices. Let us choose for the values 
of n, the time /, the direction of vector part K, of k f 
and two perpendicular directions 1, 2. We shall not 
change the expression for these two 1, 2 for these are 
represented by transverse quanta. But we must find 
OyrvYt)— (7k""7k)- Now k = kat— Ky K , where 
K= (K-K)*, and we have shown above that k replacing 
the 7 M gives zero. 23 Hence Ky^ is equivalent to & 4 Y< and 

(yr • -7<)-(7k- • ■y K ) = ((K 2 -h 2 )/K 2 )(yr • -y t ), 

so that on multiplying by k~ 2 d i k = d i k(k i 2 —K 2 )^ 1 the net 
effect is —{y t ---yt)d A k/K 2 . The y t means just scalar 
waves, that is, potentials produced by charge density. 
The fact that \/K 2 does not contain ki means that & 4 
can be integrated first, resulting in an instantaneous 
interaction, and the d z K/K 2 is just the momentum 
representation of the Coulomb potential, 1/r. 

9. KLEIN GORDON EQUATION 

The methods may be readily extended to particles of 
spin zero satisfying the Klein Gordon equation, 24 

□V- mhlf = id (A M tA)/a^+ iA ^/dx^ -A>A rf. (35) 



23 A little moi;e care is required when both 7/s act on the same 
particle. Define x = ^47(+^7K. an d consider (k- ■ -x) + (x- ■_■ k). 
Exactly this term would arise if a system, acted on by potential x 
carrying momentum — ft, is disturbed by an added potential ft of 
momentum +ft (the reversed sign of the momenta in the inter- 
mediate factors in the second term x- ■ k has no effect since we 
will later integrate over all ft). Hence as shown above the result is 
zero, but since (ft- • -x) + (x- • •fc) = W(7i- • -yt) — i^(7K - • % 7k) 
we can still conclude (7K' ■ - 7k) =k A 2 K~ 2 (yi- ■•Yt)- 

24 The equations discussed in this section were deduced from the 
formulation of the Klein Gordon equation given in reference 5, 
Section 14. The function 4> in this section has only one component 
and is not a spinor. An alternative formal method of making the 
equations valid for spin zero and also for spin i is (presumably) 
by use of the Kemmer-Dufhn matrices /? M , satisfying the commu- 
tation relation 

If we interpret a to mean a^, rather than a M 7 M . f° r an . v a .">. a ^ 
of the equations in momentum space will remain formally identical 
to those for the spin 1/2; with the exception of those in which a 
denominator (p—m)~ l has been rationalized to (J>+»i)(J> 3 — m-)~ l 
since p 2 is no longer equal to a number, p-p. But p 3 does equal 
(p-p)p so that (p— m)~ l may now be interpreted as (mp+m? 
+p t —p'.P)(p'p—m 2 )~ 1 m~ 1 . This implies that equations _ in co- 
ordinate space will be valid of the function K+{2, 1) is given as 
A' + (2, l) = [(tV 2 +m)-»r 1 (V2 2 +02 2 )}7 + (2, 1) with v^frd/dxt,.. 
This is all in virtue of the fact that the many component wave 
function \p (5 components for spin 0, 10 for spin 1) satisfies 
(iV — m)\P = A\P which is formally identical to the Dirac Equation. 
See W. Pauli, Rev. Mod. Phys. 13, 203 (1940). 



191 



782 



FEYNM AN 



The important kernel is now I+{2, 1) denned in (I, Eq. 
(32)). For a free particle, the wave function \p(2) satisfies 
+DV - m 2 $=0. At a point, 2, inside a space time region 
it is given by 

#(2)-J[*(i)az+(2,i)/a*i, 

-(d^/dx lli )I + (2,im il (l)dW u 

(as is readily shown by the usual method of demon- 
strating Green's theorem) the integral being over an 
entire 3-surface boundary of the region (with normal 
vector N,,). Only the positive frequency components of 
\f/ contribute from the surface preceding the time corre- 
sponding to 2, and only negative frequencies from the 
surface future to 2. These can be interpreted as electrons 
and positrons in direct analogy to the Dirac case. 

The right-hand side of (35) can be considered as a 
source of new waves and a series of terms written down 
to represent matrix elements for processes of increasing 
order. There is only one new point here, the term in 
AyAy by which two quanta can act at the same time. 
As an example, suppose three quanta or potentials, 
ay exp(— iq a -x), by exp(— iqb-x), and c„ exp(— iq c -x) are 
to act in that order on a particle of original momentum 
poy so that pa=po+g a and pb^pa+Qb) the final mo- 
mentum being p c = pb-\-g c - The matrix element is the 
sum of three terms (J> 2 = pypy) (illustrated in Fig. 7) 

(pc-c+py^iPb'-mT'ipyb+pa-b) 

X(pa 2 -m*)- l (pa-a+p a -a) n ,s 
-(pc-c+Pb-cW-mTKb-a) K ™ } 

-tc-b)ip a 2 -m 2 )- l (p a -a+Po-a). 

The first comes when each potential acts through the 
perturbation id(A lt \p)/dx li -\-iA ll d^/dx ll . These gradient 
operators in momentum space mean respectively the 
momentum after and before the potential Ay operates. 
The second term comes from by and a„ acting at the 
same instant and arises from the AyAy term in (a). 
Together by and ay carry momentum qby-\-q ai i so that 
after b -a operates the momentum is po-\-q a -\-Qb or p b . 
The final term comes from Cy and by operating together 
in a similar manner. The term AyAy thus permits a new 
type of process in which two quanta can be emitted (or 
absorbed, or one absorbed, one emitted) at the same 
time. There is no a ■ c term for the order a, b, c we have 
assumed. In an actual problem there would be other 
terms like (36) but with alterations in the order in 
which the quanta a, b, c act. In these terms a-c would 
appear. 

As a further example the self-energy of a particle of 
momentum py is 

(e 2 /2irim) fl(2p-k)y((p- k) 2 -m 2 )~ l 

X{2p-k)y-byy']d i kk- 2 Cih?), 

where the 5^=4 comes from the AyAy term and repre- 



sents the possibility of the simultaneous emission and 
absorption of the same virtual quantum. This integral 
without the C(k 2 ) diverges quadratically and would not 
converge if C(k 1 ) = — \ 2 /(k 2 — X 2 ). Since the interaction 
occurs through the gradients of the potential, we must 
use a stronger convergence factor, for example C(k 2 ) 
= X 4 (F-X 2 )- 2 , or in general (17) with f x X z G(\)d\ = 0. 
In this case the self-energy converges but depends 
quadratically on the cut-off X and is not necessarily 
small compared to m. The radiative corrections to 
scattering after mass renormalization are insensitive to 
the cut-off just as for the Dirac equation. 

When there are several particles one can obtain Bose 
statistics by the rule that if two processes lead to the 
same state but with two electrons exchanged, their 
amplitudes are to be added (rather than subtracted as 
for Fermi statistics). In this case equivalence to the 
second quantization treatment of Pauli and Weisskopf 
should be demonstrable in a way very much like that 
given in / (appendix) for Dirac electrons. The Bose 
statistics mean that the sign of contribution of a closed 
loop to the vacuum polarization is the opposite of what 
it is for the Fermi case (see I). It is (pb = p a +q) 



Jy, 



2 irim 



fl(P» 



+ pay){pb„ + p a ,){pa 2 -m 2 )- 



X(p b 2 -m 2 )- l -8y y (p a 2 -m 2 )- 



-by V {pb 2 -m 2 )-^ 



giving, 
j p = 



e 2 r 

-(qyq-8y V q 2 )\ 

IT L 



1 X 2 1 \m 2 -q' 

-In— +- 

6 m 2 9 3q 2 



V tan0/J 



the notation as in (33). The imaginary part for (<7 2 )*> 2m 
is again positive representing the loss in the probability 
of finding the final state to be a vacuum, associated with 
the possibilities of pair production. Fermi statistics 
would give a gain in probability (and also a charge 
renormalization of opposite sign to that expected). 




Fig. 7. Klein-Gordon particle in three potentials, Eq. (36). 
The coupling to the electromagnetic field is now, for example, 
po-a+pa-a, and a new possibility arises, (b), of simultaneous inter- 
action with two quanta a -b. The propagation factor is now 
(/>•/»— w 2 ) -1 for a particle of momentum py. 



192 



QUANTUM ELECTRODYNAMICS 



783 



10. APPLICATION TO MESON THEORIES 

The theories which have been developed to describe 
mesons and the interaction of nucleons can be easily 
expressed in the language used here. Calculations, to 
lowest order in the interactions can be made very easily 
for the various theories, but agreement with experi- 
mental results is not obtained. Most likely all of our 
present formulations are quantitatively unsatisfactory. 
We shall content ourselves therefore with a brief sum- 
mary of the methods which can be used. 

The nucleons are usually assumed to satisfy Dirac's 
equation so that the factor for propagation of a nucleon 
of momentum p is (p—M)~ l where M is the mass of the 
nucleon (which implies that nucleons can be created in 
pairs). The nucleon is then assumed to interact with 
mesons, the various theories differing in the form as- 
sumed for this interaction. 

First, we consider the case of neutral mesons. The 
theory closest to electrodynamics is the theory of vector 
mesons with vector coupling. Here the factor for emis- 
sion or absorption of a meson is gy^ when this meson is 
"polarized" in the /x direction. The factor g, the 
"mesonic charge," replaces the electric charge e. The 
amplitude for propagation of a meson of momentum q 
in intermediate states is (q 2 — /x 2 ) -1 (rather than q~~ 2 as it 
is for light) where n is the mass of the meson. The neces- 
sary integrals are made finite by convergence factors 
C(q 2 — y?) as in electrodynamics. For scalar mesons with 
scalar coupling the only change is that one replaces the 
7 M by 1 in emission and absorption. There is no longer 
a direction of polarization, /x, to sum upon. For pseudo- 
scalar mesons, pseudoscalar coupling replace 7 M by 
yb = iyxy y yzyt- For example, the self-energy matrix of 
a nucleon of momentum p in this theory is 



tf/irO/ 



7 6 (/'-ft-M)- 1 75^(fc 2 -MVC(ft 2 -M 2 ). 



Other types of meson theory result from the replace- 
ment of 7 M by other expressions (for example by 
Ky^Y" - 7/yJ with a subsequent sum over all ji and v 
for virtual mesons). Scalar mesons with vector coupling 
result from the replacement of y^ by yr x q where q is the 
final momentum of the nucleon minus its initial mo- 
mentum, that is, it is the momentum of the meson if 
absorbed, or the negative of the momentum of a meson 
emitted. As is well known, this theory with neutral 
mesons gives zero for all processes, as is proved by our 
discussion on longitudinal waves in electrodynamics. 
Pseudoscalar mesons with pseudo-vector coupling corre- 
sponds to 7 M being replaced by n~ l y b q while vector 
mesons with tensor coupling correspond to using 
(2fji)~ 1 (y IJl q—qy l j). These extra gradients involve the 
danger of producing higher divergencies for real proc- 
esses. For example, y b q gives a logarithmically divergent 
interaction of neutron and electron. 25 Although these 
divergencies can be held by strong enough convergence 

?5 M. Slotnick and W. Heitler, Phys. Rev. 75, 1645 (1949). 



factors, the results then are sensitive to the method used 
for convergence and the size of the cut-off values of X. 
For low order processes y.~ l y 5 q is equivalent to the 
pseudoscalar interaction 2Mpr l y b because if taken be- 
tween free particle wave functions of the nucleon of 
momenta pi and p-2 = pi~{-q, we have 

{ihysqui) = (H2y&(p2—pi)u\) = - (uiptynUi) 

- (U'ly-opiUi) = - 2M{il 2 y b uy) 

since 75 anticommutes with p> and pi operating on the 
state 2 equivalent to M as is pi on the state 1. This 
shows that the 75 interaction is unusually weak in the 
non-relativistic limit (for example the expected value 
of 75 for a free nucleon is zero), but since 7s 2 = 1 is not 
small, pseudoscalar theory gives a more important inter- 
action in second order than it does in first. Thus the 
pseudoscalar coupling constant should be chosen to fit 
nuclear forces including these important second order 
processes. 26 The equivalence of pseudoscalar and pseudo- 
vector coupling which holds for low order processes 
therefore does not hold when the pseudoscalar theory 
is giving its most important effects. These theories will 
therefore give quite different results in the majority of 
practical problems. 

In calculating the corrections to scattering of a nu- 
cleon by a neutral vector meson field (7^) due to the 
effects of virtual mesons, the situation is just as in 
electrodynamics, in that the result converges without 
need for a cut-off and depends only on gradients of the 
meson potential. With scalar (1) or pseudoscalar (75) 
neutral mesons the result diverges logarithmically and 
so must be cut off. The part sensitive to the cut-off, 
however, is directly proportional to the meson poten- 
tial. It may thereby be removed by a renormalization 
of mesonic charge g. After this renormalization the re- 
sults depend only on gradients of the meson potential 
and are essentially independent of cut-off. This is in 
addition to the mesonic charge renormalization coming 
from the production of virtual nucleon pairs by a meson, 
analogous to the vacuum polarization in electro- 
dynamics. But here there is a further difference from 
electrodynamics for scalar or pseudoscalar mesons in 
that the polarization also gives a term in the induced 
current proportional to the meson potential representing 
therefore an additional renormalization of the mass of 
the meson which usually depends quadratically on the 
cut-off. 

Next consider charged mesons in the absence of an 
electromagnetic field. One can introduce isotopic spin 
operators in an obvious way. (Specifically replace the 
neutral 75, say, by r t y b and sum over i=l, 2 where 
ti=t + +t_, T%=i(r+— t_) and r + changes neutron to 
proton (r+ on proton = 0) and r_ changes proton to 
neutron.) It is just as easy for practical problems simply 
to keep track of whether the particle is a proton or a 
neutron on a diagram drawn to help write down the 

26 H. A. Bethe, Bull. Am. Plus. Soc. 24, 3, Z3 (Washington, 
1949). 



193 



784 



FE YN M AN 



matrix element. This excludes certain processes. For 
example in the scattering of a negative meson from qi 
to (72 by a neutron, the meson qz must be emitted first 
(in order of operators, not time) for the neutron cannot 
absorb the negative meson q\ until it becomes a proton. 
That is, in comparison to the Klein Nishina formula (15), 
only the analogue of second term (see Fig. 5(b)) would 
appear in the scattering of negative mesons by neu- 
trons, and only the first term (Fig. 5(a)) in the neutron 
scattering of positive mesons. 

The source of mesons of a given charge is not con- 
served, for a neutron capable of emitting negative me- 
sons may (on emitting one, say) become a proton no 
longer able to do so. The proof that a perturbation q 
gives zero, discussed for longitudinal electromagnetic 
waves, fails. This has the consequence that vector me- 
sons, if represented by the interaction y^ would not 
satisfy the condition that the divergence of the poten- 
tial is zero. The interaction is to be taken 27 as y^— m~ 2 <?m<7 
in emission and as y„ in absorption if the real emission 
of mesons with a non-zero divergence of potential is to 
be avoided. (The correction term m~ 2 <7^9 gives zero in 
the neutral case.) The asymmetry in emission and ab- 
sorption is only apparent, as this is clearly the same 
thing as subtracting from the original y m - • -y^, a term 
fir 2 q- ■ q. That is, if the term —ir 2 q ll q is omitted the 
resulting theory describes a combination of mesons of 
spin one and spin zero. The spin zero mesons, coupled 
by vector coupling q, are removed by subtracting the 
term n~ 2 q- • -q. 

The two extra gradients q- • q make the problem of 
diverging integrals still more serious (for example the 
interaction between two protons corresponding to the 
exchange of two charged vector mesons depends quad- 
ratically on the cut-off if calculated in a straightforward 
way). One is tempted in this formulation to choose 
simply 7 M • • • 7 M and accept the admixture of spin zero 
mesons. But it appears that this leads in the conven- 
tional formalism to negative energies for the spin zero 
component. This shows one of the advantages of the 



27 The vector meson field potentials <p„ satisfy 

— d/dx v {dip fl /dx„—dip v /dx l i) — n 1 ip )X ——A:TrSf i , 

where s^, the source for such mesons, is the matrix element of 
7^ between states of neutron and proton. By taking the divergence 
d/dXp of both sides, conclude that d l p v /dx v = ^iry.~ i ds 1 ,/dx ll so that 
the original equation can be rewritten as 

W-<p IJL -iJ?<Pn=-'±ir{s l L+vr-d/dx ll {ds v /dx v )). 
The right hand side gives in momentum representation jy. 
— fj.~ 2 q^q v y v the left yields the (q 2 — ju 2 ) -1 and finally the interaction 
s^tpp in the Lagrangian gives the y M on absorption. 

Proceeding in this way find generally that particles of spin one 
can be represented by a four-vector u„. (which, for a free particle 
of momentum q satisfies q-u = 0). The propagation of virtual 
particles of momentum q from state v to n is represented by 
multiplication by the 4-4 matrix (or tensor) P,, v = (8^— yT^q^qv) 
X (? 2 — m 2 ) -1 - The first-order interaction (from the Proca equation) 
with an electromagnetic potential a exp( — ik-x) corresponds to 
multiplication by the matrix E^=(q 2 -a-\-qi-a)S liv — q2va.fi— qi^a v 
where q\ and _q-i = q\+k are the momenta before and after the 
interaction. Finally, two potentials a, b may act simultaneously, 
with matrix E' flv = — (a-b)du V +bna y . 



method of second quantization of meson fields over the 
present formulation. There such errors of sign are obvi- 
ous while here we seem to be able to write seemingly 
innocent expressions which can give absurd results. 
Pseudovector mesons with pseudovector coupling corre- 
spond to using 75(7^- M~ 2 (M7) for absorption and 7=,7 M 
for emission for both charged and neutral mesons. 

In the presence of an electromagnetic field, whenever 
the nucleon is a proton it interacts with the field in the 
way described for electrons. The meson interacts in the 
scalar or pseudoscalar case as a particle obeying the 
Klein-Gordon equation. It is important here to use the 
method of calculation of Bethe and Pauli, that is, a 
virtual meson is assumed to have the same "mass" dur- 
ing all its interactions with the electromagnetic field. 
The result for mass n and for (/x 2 +X 2 ) 5 are subtracted 
and the difference integrated over the function G(\)d\. 
A separate convergence factor is not provided for each 
meson propagation between electromagnetic interac- 
tions, otherwise gauge invariance is not insured. When 
the coupling involves a gradient, such as y^q where q is 
the final minus the initial momentum of the nucleon, 
the vector potential A must be subtracted from the 
momentum of the proton. That is, there is an additional 
coupling ±75-4 (plus when going from proton to neu- 
tron, minus for the reverse) representing the new possi- 
bility of a simultaneous emission (or absorption) of 
meson and photon. 

Emission of positive or absorption of negative virtual 
mesons are represented in the same term, the sign of the 
charge being determined by temporal relations as for 
electrons and positrons. 

Calculations are very easily carried out in this way 
to lowest order in g 2 for the various theories for nucleon 
interaction, scattering of mesons by nucleons, meson 
production by nuclear collisions and by gamma-rays, 
nuclear magnetic moments, neutron electron scattering, 
etc., However, no good agreement with experiment re- 
sults, when these are available, is obtained. Probably 
all of the formulations are incorrect. An uncertainty 
arises since the calculations are only to first order in g 2 , 
and are not valid if g 2 /hc is large. 

The author is particularly indebted to Professor H. 
A. Bethe for his explanation of a method of obtaining 
finite and gauge invariant results for the problem of 
vacuum polarization. He is also grateful for Professor 
Bethe's criticisms of the manuscript, and for innumer- 
able discussions during the development \>i this work. 
He wishes to thank Professor J. Ashkin for his careful 
reading of the manuscript. 

APPENDIX 

In this appendix a method will be illustrated by which the 
simpler integrals appearing in problems in electrodynamics can 
be directly .evaluated. The integrals arising in more complex 
processes lead to rather complicated functions, but the study of 
the relations of one integral to another and their expression in 
terms of simpler integrals may be facilitated by the methods 
given here. 



194 



QUANTUM ELECTRODYNAMICS 



785 



As a typical problem consider the integral (12) appearing in 
the first order radiationless scattering problem: 



fy»(Pt 



i)-^a(pi-k-m)- 1 y lt Jr^d*kC(k t ), (la) 



where we shall take C(k 2 ) to be typically -X*(**-X 8 ) -1 and 
d 4 k means (2Tr)~ 2 dkidk2dk i dk i . We first rationalize the factors 
(p-k-m)- 1 =(J>-k+m)((J>-ky-m 2 )- i obtaining, 



/ 



y^-k+t^aiPi-k+f^y^d'kCik 2 ) 

X((/>.-ft) 2 -m 2 )- 1 ((/'2-ft) 2 -w 2 )- 1 . (2a) 

The matrix expression may be simplified. It appears to be best to 
do so after the integrations are performed. Since AB=2A B — BA 
where A-B = A M B M is a number commuting with all matrices, find, 
if R is any expression, and A a vector, since y^A= — Ay ll +2A I1 , 



y^ARy ll = -AypRyp+lRA. 



(3a) 



Expressions between two 7 M 's can be thereby reduced by induc- 
tion. Particularly useful are 



y lx Ay IJ ,= -2A 
y ll ABy ll = 2(AB+BA)- 
y„ABCy lx =-2CBA 



AA-B 



(4a) 



where A, B, C are any three vector-matrices (i.e., linear com- 
binations of the four 7's). 

In order to calculate the integral in (2a) the integral may be 
written as the sum of three terms (since k = k a y„), 

yvU>2+m)a(pi+m)y ti Ji-[y ll y a a(j) l +m)y ll 

+ y^{p2+m)ayay^V2+ynyaay T y^Ji, (5a) 
where 

/ (1 ; 8ig )=J*(i ; ;& ff ;jW r )fc-wfec(ft*) 

X ((fit- k)*-m*)-i((fii - ky-m*)- 1 . (6a) 

That is for Ji the (1; k„\ k„k T ) is replaced by 1, for Ji by k a , and 
for 7 3 by k a k T . 

More complex processes of the first order involve more factors 
like ((pa— k) 2 — m 2 )~ l and a corresponding increase in the number 
of k's which may appear in the numerator, as k„k T k v - ■ ■ . Higher 
order processes involving two or more virtual quanta involve 
similar integrals but with factors possibly involving k+k' instead 
of just k, and the integral extending on k- 2 d 4 kC(k 2 )k'- 2 d 4 k'C(k' 2 ). 
They can be simplified by methods analogous to those used on 
the first order integrals. 

The factors (p — k) 2 — m 2 may be written 



(p-k) 2 -tn 2 =k 2 -2p-k-A, 



(7a) 



where A = m 2 —p 2 , A l = mi i —p l 2 , etc., and we can consider dealing 
with cases of greater generality in that the different denominators 
need not have the same value of the mass m. In our specific prob- 
lem (6a.),pi 2 = m 2 so that Ai = 0, but we desire to work with greater 
generality. 

Now for the factor C(k 2 )/k 2 we shall use -X 2 (fc 2 -X 2 )->fc- 2 . 
This can be written as 



-X 2 /(* 2 -X 2 )ft 2 = k~ 2 C(k 2 ) = - f X2 dL(k 2 -L)- 
J 



(8a) 



Thus we can replace k~ 2 C(k 2 ) by (k 2 -L)~ 2 and at the end inte- 
grate the result with respect to L from zero to X 2 . We can for 
many practical purposes consider X 2 very large relative to m 2 or/) 2 . 
When the original integral converges even without the con- 
vergence factor, it will be obvious since the L integration will then 
be convergent to infinity. If an infra-red catastrophe exists in the 
integral one can simply assume quanta have a small mass X m j n 
and extend the integral on L from X 2 min to X 2 , rather than from 
zero to X s . 



We then have to do integrals of the form 

j{\; k„; k a k T )d<k(k 2 -L)^(k 2 -2p v k-A 1 )-i 

X(k 2 -2p 2 -k-A«)~\ (9a) 

where by (1; k„; k„k T ) we mean that in the place of this symbol 
either 1, or k a , or k„k T may stand in different cases. In more 
complicated problems there may be more factors (k 2 — 2pi-k — A i )~ 1 
or other powers of these factors (the (k 2 — Z.) -2 may be considered 
as a special case of such a factor with Pi = 0, Ai = L) and further 
factors like k a k T k p - • ■ in the numerator. The poles in all the factors 
are made definite by the assumption that L, and the A's have 
infinitesimal negative imaginary parts. 

We shall do the integrals of successive complexity by induction. 
We start with the simplest convergent one, and show 



/' 



d<k(k 2 -L)- 3 =(8iL)- 



(10a) 



For this integral is f(2Tr)-^dk i d 3 K(k i 2 -K- K-L)~ 3 where the 
vector K, of magnitude AT=(K-K)* is k\, k«_, k 3 . The integral on 
ki shows third order poles at k i = + (K 2 +L)i and k<= -(K 2 +L)i. 
Imagining, in accordance with our definitions, that L has a small 
negative imaginary part only the first is below the real axis. The 
contour can be closed by an infinite semi-circle below this axis, 
without change of the value of the integral since the contribution 
from the semi-circle vanishes in the limit. Thus the contour can 
be shrunk about the pole k i = + (K 2 +L) i and the resulting & 4 inte- 
gral is — 27ri times the residue at this pole. Writing k i ={K 2 -\-L)^-\-e 
and expanding (k i 2 -K 2 -L)~ 3 =e- 3 (t+2(K 2 +L)i)- 3 in powers of 
e, the residue, being the coefficient of the term e _1 , is seen to be 
6(2(K 2 + L)*)~ b so our integral is 

- (3i/32ir)J'~ 47rXVA'(ii?-r-Z.)- 5 ' 2 = (3/8»')(l/3I) 

establishing (10a). 

We also have J"k a d i k(k 2 — L)- 3 = from the symmetry in the 
k space. We write these results as 

(11a) 



(Si)j'(l; k,)d*k{k 2 -L)- 3 = (1; O)!,- 1 , 

where in the brackets (1; k a ) and (1; 0) corresponding entries are 
to be used. 
Substituting k= k'—p in (11a), and calling L—p 2 =A shows that 

(S,i)j\l; k,)d<k(k 2 -2p-k-A)- 3 =(\; p^tJt+A)-*. (12a) 

By differentiating both sides of (12a) with respect to A, or with 
respect to p T there follows directly 

(24i) f(U k,; k a k T )d i k(k 2 -2p-k-A)-* 

= -(1; p„; p„Pr-^ar(J> 2 +^))(P 2 +^)^- (13a) 

Further differentiations give directly successive integrals in- 
cluding more k factors in the numerator and higher powers of 
(k 2 — 2p-k — A) in the denominator. 

The integrals so far only contain one factor in the denominator. 
To obtain results for two factors we make use of the identity 



a- l b~ l = f* dx(ax+b(l-x))-*, 



(14a) 



(suggested by some work of Schwinger's involving Gaussian inte- 
grals). This represents the product of two reciprocals as a para- 
metric integral over one and will therefore permit integrals with 
two factors to be expressed in terms of one. For other powers of 
a, b, we make use of all of the identities, such as 



(15a) 



a -*b- l =y 2xdx(ax+b(\-x))- 3 , 
(14a) by successive differentiatk 
a integral, such as 
i)f(l;k a )d i k(k 2 -2p l -k-Ai)- 2 (k 2 -2p 2 -k-A 2 )- i , (16a) 



deducible from (14a) by successive differentiations with respect 
to a or b. 
To perform an integral, such as 



195 



786 



k 



FEYNMAN 



write, using (15a), 

{k 2 -2p v k-A l )- 2 {k 2 -2p 2 -k-A 2 )^ = £ 2xdx{k 2 -2p x -k-A x )^, 

where 

px = xPi+(l-x)p 2 and A z = xA 1 -\-(l-x)A 2 , (17a) 

(note that A z is not equal to m 2 — pi 1 ) so that the expression (16a) 
is (8i)f i 2xdxf(l;k«)d i k(k 2 -2p z -k-A x )- :i which may now be 
evaluated by (12a) and is 



(16a) = £ {\;p xa )2xdx(p x 2 +A z )~\ 



(18a) 



where p x , A x are given in (17a). The integral in (18a) is elementary, 
being the integral of ratio of polynomials, the denominator of 
second degree in x. The general expression although readily ob- 
tained is a rather complicated combination of roots and logarithms. 
Other integrals can be obtained again by parametric differentia- 
tion. For example differentiation of (16a), (18a) with respect to 
A 2 or p2 T gives 

(8i)f(l;k a ; k < ,k T )d i k(k 2 -2p r k-Ai)- i (k i -2p2-k-A2)- 2 

X2x(l-x)dx(J> x *+A x )-*, (19a) 

again leading to elementary integrals. 

As an example, consider the case that the second factor is just 
(k 2 -L)^ and in the first put Pi=p, A 1 = A. Then p x = xp, 
A z = xA+(l — x)L. There results 

(Si)f(l;k a ; k a k T )d i k{k 2 -L)- 2 {,k 2 -2p-k-A)^ 

= -£ (liXP^xtprPr-farixtp+At)) 

X2x(l-x)dx(x*p*+A x )- i . (20a) 

Integrals with three factors can be reduced to those involving 
two by using (14a) again. They, therefore, lead to integrals with 
two parameters (e.g., see application to radiative correction to 
scattering below). 

The methods of calculation given in this paper are deceptively 
simple when applied to the lower order processes. For processes 
of increasingly higher orders the complexity and difficulty in- 
creases rapidly, and these methods soon become impractical in 
their present form. 

A. Self-Energy 

The self-energy integral (19) is 

(e'/TrDfy^ip-k-m^y^d^kCik 2 ), (19) 

so that it requires that we find (using the principle of (8a)) the 
integral on L from to X 2 of 

fy li (p-k+m)y ll d , k(k 2 -L)- 2 (k 2 -2p-k)- 1 , 

since {p-k) 2 -m 2 =k 2 -2p-k, SLSp 2 =m 2 . This is of the form (16a) 
with Ai = L, />i = 0, A 2 = 0, p2=p so that (18a) gives, since 
p x =(l-x)p, A z = xL, 

(8») J*(l ; k <7 )d*k(k 2 -L)^(k 2 -2p ■ k)~ l 

= £ (l;(l-x)p a )2xdx((l-x)W+xL)-\ 

or performing the integral on L, as in (8), 

(8i)/(l; K)d^kk^C{k 2 ){k 2 -2p-k)^ 

Jo ,r " (l-x) 2 m 2 

Assuming now that \ 2 »m 2 we neglect (1— x)hn 2 relative to 
xX 2 in the argument of the logarithm, which then becomes 
(X 2 /m 2 )(x/(l-x) 2 ). Then since fHx ln(*(l -*)"*) = 1 and 



f a l (l-x)dx ln(x(l-x)- 2 )= -(1/4) find 
(8i)f(l;k <r )k- i C(k i )d l k(k 2 -2p-kr i 

so that substitution into (19) (after the (p-k-m)' 1 in (19) is 
replaced by (p—k-\-m)(k 2 — 2p-k)~ 1 ) gives 



(19) = ( e 2 /8^)T,[^+w)(21n(X 2 A« 2 ) + 2) 

-/»(ln(X 2 /m 2 )-i)] 7M 
= (e 2 /8 1 r)[8w(ln(X 2 /« 2 )+l)-/>(2 1n(X 2 /w 2 ) + 5)], 



(20) 



using (4a) to remove the 7^'s. This agrees with Eq. (20) of the text, 
and gives the self-energy (21) when p is replaced by m. 

B. Corrections to Scattering 

The term (12) in the radiationless scattering, after rationalizing 
the matrix denominators and using pi 2 = p 2 2 = m 2 requires the 
integrals (9a), as we have discussed. This is an integral with 
three denominators which we do in two stages. First the factors 
(k 2 — 2pi-k) and (k 2 — 2p 2 -k) are combined by a parameter y; 

{k 2 -2p v k)-\k 2 -2p 2 -k)^ = £ dy(k 2 -2p y -k)^, 
from (14a) where 

Pu=ypi+i\~y)p2- (2ia) 

We therefore need the integrals 

(M)/(l; V; k„k T )d i k{k 2 -L)~ 2 (k 2 -2p y -k)- Ji , (22a) 

which we will then integrate with respect to y from to 1. Next 
we do the integrals (22a) immediately from (20a) with p =p y , A = : 

(22a) = -£ £ (1; xp y ,; x 2 p ya p yT 

-hd, T (x 2 p y 2 +(l-x)L))2x(l-x)dx(x 2 p y 2 +L(l-x))- 2 dy. 

We now turn to the integrals on L as required in (8a). The first 
term, (1), in (1; k„; k a k T ) gives no trouble for large L, but if L 
is put equal to zero there results x~~ 2 p y ~ 2 which leads to a diverging 
integral on x as x—*0. This infra-red catastrophe is analyzed by 
using Xmin 2 for the lower limit of the L integral. For the last term 
the upper limit of L must be kept as X 2 . Assuming X ro , n 2 <3C/> y 2 <3C X* 
the x integrals which remain are trivial, as in the self-energy case. 
One finds 

- (U)J{k 2 - X min 2 )-W *kC(k 2 - X min 2 ) (k 2 - 2p x ■ kT\k 2 - 2p2 ■ k)~ l 

= £p y -*dy\n(p y 2 /\ miD 2 ) (23a) 

-(8i)fk a k- 2 d i kC(k 2 )(k 2 -2p l -k)- 1 (k 2 -2p2-k)-i 

= l£ P».P*-*dy, (24a) 
-(8i)fk a k T k-WkC(k 2 )(k 2 -2p 1 -k)-Kk 2 -2p2-kr i 

= £ PvPvTpv-Vy-farf* dy InOtyT 8 ) + * 5 "- (25a) 
The integrals on y give, 
£ p y ~*dy IntaAmin- 2 ) = 4(m 2 sin20)-i[~0 l n (wX mi »- 1 ) 

— f ata.nada\, (26a) 
£ Pv<rPv~ 2 dy = 8(m i sin20)- 1 (J>i ff +£2„), (27a) 

£ PycP V rPy- i dy = e(2m 2 An29)-^{pi 9 +p lT )(pu+ptr) 

+«^V?T(l-0ctn0), (28a) 

£ dy ln(X 2 A,- 2 ) = ln(X7w 2 )+2(l-0ctn0). (29a) 



196 



QUANTUM ELECTRODYNAMICS 



787 



These integrals on y were performed as follows. Since p2=p\+q 
where q is the momentum carried by the potential, it follows from 
p 2 2=p l t=m? that 2pi-q=—q 2 so that since py=pi+q(l — y), 
p u 2 = m 2 —q 2 y(l — y). The substitution 2y— l = tana:/tan0 where 6 
is defined by 4»z 2 sin 2 = <7 2 is useful for it means p v 2 = m 2 sec 2 a/sec 2 d 
and p v ~Hy= (m 2 sin20) _1 da! where a. goes from —6 to -f 0. 

These results are substituted into the original scattering formula 

(2a), giving (22). It has been simplified by frequent use of the 

fact that pi operating on the initial state is m, and likewise pi 

when it appears at the left is replacable by m. (Thus, to simplify: 

y^ptapiy li = -Ipidpt by (4a), 

= -2(p 2 -q)a(J) l +q) = -2(m-q)a(m+q). 
A term like qaq= — q 2 a-\-2{a-q)q is equivalent to just — q 2 a since 
q=p 2 —pi = m — m has zero matrix element.) The renormalization 
term requires the corresponding integrals for the special case 
9 = 0. 

C. Vacuum Polarization 

The expressions (32) and (32') for / M „ in the vacuum polariza- 
tion problem require the calculation of the integral 

/„,(>»-) = --. fsply ll (p-^q^m)y v ip+iq+m)2d i p 

X{{p-hqT~-m 2 )-KiP+\qY-ni 2 r\ (32) 
where we have replaced p by p—\q to simplify the calculation 
somewhat. We shall indicate the method of calculation by studying 
the integral, 



/( IW 2)= J p.p T d*p(U>-lq)*-m*)-K(P+iqV 



The factors in the denominator, p 2 — p ■ q— m 2 -\-\q 2 and p 2J rp-q 
— m 2 -\-\q 2 are combined as usual by (8a) but for symmetry we 
substitute .v=j(1 + ij), (1— x) = i(l — rf) and integrate jj from 
-1 to +1: 

I{m 2 ) =f2 p < ,p r d>p(fi 2 -r ) p-q-m 2 +\q 2 )- 2 dr l j2. (30a) 

But the integral on p will not be found in our list for it is badly 
divergent. However, as discussed in Section 7, Eq. (32') we do not 
wish I(m 2 ) but rather /o™[/(m 2 )-/(w 2 +X 2 )]G(X)JX. We can 
calculate the difference I{m 2 ) — I(m 2 -\-\ 2 ) by first calculating the 
derivative I'(m 2 +L) of / with respect to m 2 at m 2 -\-L and later 
integrating L from zero to X 2 . By differentiating (30a), with 
respect to m 2 find, 

I'(m 2 +L)= f_* p a p r d*p{p 2 --op-q- m 2 - L+\q 2 THr,. 

This still diverges, but we can differentiate again to get 

I"(m 2 +L)=?>f* l i p G p T d i p{p 2 -r l p-q-m 2 -L+\q 2 )- i dr 1 



-(8i) -1 XT (.h%qrD-*-$5,, T iri)dv 



(31a) 



(where Z? = j(tj 2 — \)q 2 +m 2 -\-L), which now converges and has been 
evaluated by (13a) with p=\riq and A — m 2 +L— \q 2 . Now to get 
/' we may integrate /" with respect to L as an indefinite integral 
and we may choose any convenient arbitrary constant. This is because 
a constant C in /' will mean a term — CX 2 in I(m 2 ) — 7(w 2 +X 2 ) 
which vanishes since we will integrate the results times G(\)d\ 
and Jo'°\ 2 G(\)d\ = 0. This means that the logarithm appearing on 
integrating L in (31a) presents no problem. We may take 

I'(m 2 +L) = (Si)" 1 /* 1 lWq,qrD- l +i8, T \nDy n +C&. T , 

a subsequent integral on L and finally on 77 presents no new 
problems. There results 

- mfp.Prd*pi(p- \q) 2 ~ m*)-K(P+\q)*- m 2 )~ l 

p 4m 2 - q 2 / B \, , \ 2 1 

+ 5„r[(X 2 + w 2 )ln(X 2 ,«- 2 + l)-C'X 2 ], (32a) 



where we assume X 2 ^>w 2 and have put some terms into the arbi- 
trary constant C which is independent of X 2 (but in principle could 
depend on q 2 ) and which drops out in the integral on G{\)d\. We 
have set q 2 = 4w 2 sin 2 0. 

In a very similar way the integral with m 2 in the numerator can 
be worked out. It is, of course, necessary to differentiate this m 2 
also when calculating /' and /". There results 

-(8i)fm 2 d<p((p-iq) 2 -m 2 ri((p+lq) 2 - m 2 )-i 

= 4m 2 (l-e>ctn9)-g 2 /3+2(X 2 +w 2 )ln(X 2 m- 2 + l)-C"X 2 ), (33a) 

with another unimportant constant C". The complete problem re- 
quires the further integral, 

-(.8i)f(UP*)d i P((p-lq) 2 --m*)-K(P+$q) i -mT 1 

= (l,O)(4(l-0ctn0) + 2 1n(X 2 w- 2 )). (34a) 
The value of the integral (34a) times m 2 differs from (33a), of 
course, because the results on the right are not actually the inte- 
grals on the left, but rather equal their actual value minus their 
value for ;w 2 =»z 2 +X 2 . 

Combining these quantities, as required by (32), dropping the 
constants C, C" and evaluating the spur gives (33). The spurs are 
evaluated in the usual way, noting that the spur of any odd 
number of 7 matrices vanishes and Sp(AB)=Sp(BA) for arbi- 
trary A, B. The 57>(1)=4 and we also have 

iSpl(pi+m l ){p2-m2)l= pi- pt-mum, (35a) 

\Spl{p l +m l ){p2-m 2 )iJ> 3 +m i )(p i -m i )2 

= (pr p2—mim2)(p3- pi-m 3 m t ) 
— (Pi-p3-mim 3 )(p2-pi—m 2 m i ) 

+ (.pi-pi—mmt)(p2-pa—tnim 3 ), (36a) 
where pi, nu are arbitrary four-vectors and constants. 

It is interesting that the terms of order X 2 lnX 2 go out, so that 
the charge renormalization depends only logarithmically on X 2 . 
This is not true for some of the meson theories. Electrodynamics 
is suspiciously unique in the mildness of its divergence. 

D. More Complex Problems 

Matrix elements for complex problems can be set up in a 
manner analogous to that used for the simpler cases. We give 
three illustrations; higher order corrections to the Miller scatter- 




Fig. 8. The interaction between two electrons t< 
One adds the contribution of every figure involvi 
quanta, Appendix D. 



197 



788 



FEYNMAN 



ing, to the Compton scattering, and the interaction of a neutron 
with an electromagnetic field. 

For the M011er scattering, consider two electrons, one in state 
Mi of momentum pi and the other in state uz of momentum pi. 
Later they are found in states z< 3 , Pz and u±, p t . This may happen 
(first order in e 2 /hc) because they exchange a quantum of momen- 
tum q=Pi—pi=pi—p2 in the manner of Eq. (4) and Fig. 1. The 
matrix element for this process is proportional to (translating (4) 
to momentum space) 

(«47m«2)(«»Ym m i)? -2 - (37a) 

We shall discuss corrections to (37a) to the next order in e 2 /hc. 
(There is also the possibility that it is the electron at 2 which 
finally arrives at 3, the electron at 1 going to 4 through the ex- 
change of quantum of momentum p 3 —p2. The amplitude for this 
process, (ufY l iUi)(ii 3 y ll U2)(j>3—p2)~ 2 , must be subtracted from 
(37a) in accordance with the exclusion principle. A similar situa- 
tion exists to each order so that we need consider in detail only 
the corrections to (37a), reserving to the last the subtraction of 
the same terms with 3, 4 exchanged.) 

One reason that (37a) is modified is that two quanta may be 
exchanged, in the manner of Fig. 8a. The total matrix element 
for all exchanges of this type is 

(e?/wi)J (uiyvipi- k- m)- 1 y lt ui)(u i y v (j)2+ k— m)-^^) 

■k-'iq-kyWk, (38a) 
as is clear from the figure and the general rule that electrons of 
momentum p contribute in amplitude (p—mY 1 between inter- 
actions t m , and that quanta of momentum k contribute ft -2 . In 
integrating on d*k and summing over fi and v. we add all alterna- 
tives of the type of Fig. 8a. If the time of absorption, y^, of the 
quantum k by electron 2 is later than the absorption, y„, of q— k, 
this corresponds to the virtual state pz+k being a positron (so 
that (38a) contains over thirty terms of the conventional method 
of analysis). 

In integrating over all these alternatives we have considered all 
possible distortions of Fig. 8a which preserve the order of events 
along the trajectories. We have not included the possibilities 
corresponding to Fig. 8b, however. Their contribution is 



means that one adds with equal weight the integrals corresponding 
to each topologically distinct figure. 

To this same order there are also the possibilities of Fig. 8d 
which give 



(*/*i)f{u*Yp(Pi- k-m)-^^) 



Xiua^+q-k-my^udk-Kq-krWk, (39a) 

as is readily verified by labeling the diagram. The contributions of 
all possible ways that an event can occur are to be added. This 




w* 



I) J (uwvipz—k—m) 1 y ll .{pi — h—m) l y v Ui) 



This integral on k will be seen to be precisely the integral (12) for 
the radiative corrections to scattering, which we have worked out. 
The term may be combined with the renormalization terms result- 
ing from the difference of the effects of mass change and the terms, 
Figs. 8f and 8g. Figures 8e, 8h, and 8i are similarly analyzed. 

Finally the term Fig. 8c is clearly related to our vacuum 
polarization problem, and when integrated gives a term propor- 
tional to (u4y»U2)(u i y v Ui)J llv q~ i . If the charge is renormalized the 
term ln(X/ra) in J ^ in (33) is omitted so there is no remaining 
dependence on the cut-off. 

The only new integrals we require are the convergent integrals 
(38a) and (39a). They can be simplified by rationalizing the de- 
nominators and combining them by (14a). For example (38a) in- 
volves the factors (k 2 -2p v k)~ 1 {k 2 +2p2- k)- 1 k~ 2 (q 2 + k 2 -2q- k)~ 2 . 
The first two may be combined by (14a) with a parameter x, and 
the second pair by an expression obtained by differentiation (15a) 
with respect to b and calling the parameter y. There results a 
factor (k 2 -2p x - k)~ 2 (k 2 +yq 2 —2yq- k)' 4 so that the integrals on 
d 4 k now involve two factors and can be performed by the methods 
given earlier in the appendix. The subsequent integrals on the 
parameters x and y are complicated and have not been worked out 
in detail. 

Working with charged mesons there is often a considerable re- 
duction of the number of terms. For example, for the interaction 
between protons resulting from the exchange of two mesons only 
the term corresponding to Fig. 8b remains. Term 8a, for example, 
is impossible, for if the first proton emits a positive meson the 
second cannot absorb it directly for only neutrons can absorb 
positive mesons. 

As a second example, consider the radiative correction to the 
Compton scattering. As seen from Eq. (15) and Fig. 5 this scatter- 
ing is represented by two terms, so that we can consider the cor- 
rections to each one separately. Figure 9 shows the types of terms 
arising from corrections to the term of Fig. 5a. Calling k the 
momentum of the virtual quantum, Fig. 9a gives an integral 

fy„(p2- k- ?n)- 1 e 2 (pi+q i - k-m^e^-k- m^y^lrWk, 

convergent without cut-off and reducible by the methods outlined 
in this appendix. 

The other terms are relatively easy to evaluate. Terms b and c 
of Fig. 9 are closely related to radiative corrections (although 
somewhat more difficult to evaluate, for one of the states is not 
that of a free electron, (J>i-\-q) 2 ^m 2 ). Terms e, f are renormaliza- 
tion terms. From term d must be subtracted explicitly the effect 
of mass Am, as analyzed in Eqs. (26) and (27) leading to (28) 
with p'=pi+q, a = e2, b = e\. Terms g, h give zero since the 
vacuum polarization has zero effect on free light quanta, q\ 2 = Q, 
q2 2 = 0. The total is insensitive to the cut-off X. 

The result shows an infra-red catastrophe, the largest part 
of the effect. When cut-off at X m j n , the effect proportional to 
ln(m/X m i n ) goes as 

(e 2 /w) ln(m/X min )(l-20 ctn20), (40a) 

times the uncorrected amplitude, where (/> 2 — />0 2 = 4;n 2 sin 2 0. This 
is the same as for the radiative correction to scattering for a 
deflection pi—p\. This is physically clear since the long wave 
quanta are not effected by short-lived intermediate states. The 
infra-red effects arise 28 from a final adjustment of the field from 
the asymptotic coulomb field characteristic of the electron of 



Fig. 9. Radiative correction to the Compton scattering term 
(a) of Fig. 5. Appendix D. 



F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937). 



198 






mm^ 



QUANTUM ELECTRODYNAMICS 



789 



momentum pi before the collision to that characteristic of an 
electron moving in a new direction pi after the collision. 

The complete expression for the correction is a very complicated 
expression involving transcendental integrals. 

As a final example we consider the interaction of a neutron with 
an electromagnetic field in virtue of the fact that the neutron may 
emit a virtual negative meson. We choose the example of pseudo- 
scalar mesons with pseudovector coupling. The change in ampli- 
tude due to an electromagnetic field A = aexp(—iq-x) determines 
the scattering of a neutron by such a field. In the limit of small q 
it will vary as qa—aq which represents the interaction of a par- 
ticle possessing a magnetic moment. The first-order interaction 
between an electron and a neutron is given by the same calculation 
by considering the exchange of a quantum between the electron 
and the nucleon. In this case a M is q~ 2 times the matrix element of 
7 M between the initial and final states of the electron, the states 
differing in momentum by q. 

The interaction may occur because the neutron of momentum 
pi emits a negative meson becoming a proton which proton inter- 
acts with the field and then reabsorbs the meson (Fig. 10a). The 
matrix for this process is (J>2=pi+q), 



f(y s k)(p i -k-M)- 1 a(Pi-k-M)-Kysk)(k?-v?)-Wk. 



(41a) 



Alternatively it may be the meson which interacts with the field. 
We assume that it does this in the manner of a scalar potential 
satisfying the Klein Gordon Eq. (35), (Fig. 10b) 



■f(yM(J>i-ki-M)-Ky>ki)(W-S)-i 

Xih-a+ki-a)^ 



TWki, (42a) 



where we have put ki=ki+q. The change in sign arises because 
the virtual meson is negative. Finally there are two terms arising 
from the y b a part of the pseudovector coupling (Figs. 10c, lOd) 



J'iyMiPi-k-Myiiy^iW-SrWk, 



(43a) 



and 



fiy^iPi-k-Mr^yskKV-SrWk. (44a) 

Using convergence factors in the manner discussed in the section 
on meson theories each integral can be evaluated and the results 
combined. Expanded in powers of q the first term gives the mag- 
netic moment of the neutron and is insensitive to the cut-off, the 
next gives the scattering amplitude of slow electrons on neutrons, 
and depends logarithmically on the cut-off. 

The expressions may be simplified and combined somewhat 
before integration. This makes the integrals a little easier and also 
shows the relation to the case of pseudoscalar coupling. For 
example in (41a) the final ysk can be written as ys(k—pi+M) 
since pi = M when operating on the initial neutron state. This is 




^5*2 

p-k 






P. c. 



Fig. 10. According to the meson theory a neutron interacts with 
an electromagnetic potential a by first emitting a virtual charged 
meson. The figure illustrates the case for a pseudoscalar meson 
with pseudovector coupling. Appendix D. 

(Pi— k— M)y b +2Mys since 75 anticommutes with pi and k. The 
first term cancels the (pi—k—M) -1 and gives a term which just 
cancels (43a). In a like manner the leading factor y b k in (41a) is 
written as — 2Myi— 7s(/>2— k— M), the second term leading to a 
simpler term containing no (pi—k — M)' 1 factor and combining 
with a similar one from (44a). One simplifies the 7 5 fci and 75^2 
in (42a) in an analogous way. There finally results terms like 
(41a), (42a) but with pseudoscalar coupling 2M75 instead of 
75ft, no terms like (43a) or (44a) and a remainder, representing 
the difference in effects of pseudovector and pseudoscalar coupling. 
The pseudoscalar terms do not depend sensitively on the cut-off, 
but the difference term depends on it logarithmically. The differ- 
ence term affects the electron-neutron interaction but not the 
magnetic moment of the neutron. 

Interaction of a proton with an electromagnetic potential can 
be similarly analyzed. There is an effect of virtual mesons on the 
electromagnetic properties of the proton even in the case that the 
mesons are neutral. It is analogous to the radiative corrections to 
the scattering of electrons due to virtual photons. The sum of the 
magnetic moments of neutron and proton for charged mesons is 
the same as the proton moment calculated for the corresponding 
neutral mesons. In fact it is readily seen by comparing diagrams, 
that for arbitrary q, the scattering matrix to first order in the 
electromagnetic potential for a proton according to neutral meson 
theory is equal, if the mesons were charged, to the sum of the 
matrix for a neutron and the matrix for a proton. This is true, for 
any type or mixtures of meson coupling, to all orders in the 
coupling (neglecting the mass difference of neutron and proton). 






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